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THE CULMINATIOK
OF THE
SCIENCE OF LOGIC
WITH SYNOPSES OF
A.LL POSSIBLE VALID FORMS OF CATEGORICAL REASONING IN
SYLLOGISMS OF BOTH THR6e AND FOUR TERMS.
BY
JOHN C. SMITH,
A MEMBER OP THE BROOKLYN BAR.
- OF THE
PUBLISHED BY
HERBERT C. SMITH,
16 COURT STREET, BROOKLYN. N. Y.
Copyright, 1888, by Herbert C. Smith.
>*7//j
Electrotyped by R. Harmer Smith & Sons. 82 Beelsiiian St.. New YorK.
PREFACE.
The following are two chapters of a treatise now in
course of preparation, and to be entitled ''Logic as a
Pure Science, illustrated only by means of symbols indefi-
nite in material, but definite in logical signification, with,
synopses of all possible valid forms of categorical reason-
ing in syllogisms of both three and four terms."
The preparation of the treatise was undertaken with
but little expectation that it, or any part of it, would
ever be published ; and certainly, with no thought of its
resulting in any new contribution to the science.
The author had long thought an elementary treatise
on Logic as a pure science, with matter wholly elimi-
nated, a desideratum ; and if any such has ever appeared,
he is not aware of it. He acknowledges, however, that
his acquaintance with the literature of the science is
very limited. In writing the treatise, therefore, no con-
crete examples were employed, but only those with sym-
bols indefinite as to matter, but made definite as to
their logical signification.
The symbols adopted w^ere the letters N, D, and J, to
IV PREFACE.
represent the Minor, Middle and Major terms of the Syl-
logism ; they being the middle letters of these words
respectively. S, M and P are usually employed, as the
initials of Subject, Middle and Predicate, but S and P
are objectionable, being equally applicable to the sub-
ject and predicate of the premises (as propositions), in
each of which but one occurs in the statement of Syllo-
gisms, and that one in its appropriate place in such
representation in both premises, only in Syllogisms in
the first figure ; in one premise only, in the second and
third figures ; and in neither, in the fourth ; and their
dual possible representations tend to confusion. Dis-
tribution and non-distribution are signified by the use
of capitals to represent terms distributed, and small
letters, terms not distributed. JSTegation, in universal
propositions, is indicated by crossing the capital let-
ter representing the subject. The copula is expressed
by the characters, ^' — " for "is," and "--" for
''is not."
In translating the symbols and characters as em-
ployed in propositions into spoken language, the sig-
nification of the symbols should of course be expressed
in respect to the subject, but implied in respect to the
predicate, according to common usage and the well-
known rules that all universal propositions (and no par-
ticular) distribute the subject, and all negative (and no
afiirmative) the predicate.
PREFACE. V
Thus the four propositions, A, E, I, O, when written
in symbols and characters as above, should be read and
understood as follows :
(A) D — j All D is j {meaning All D is some J)
(E) &— J No D is J ( '' No D is «W2/ J)
(I) d — j Some D is j ( '' Some D is some J)
(0) d — ^J Some D is not J ( ^' Some D is not awy J)
The consideration of Hypotheticals was reached in
the preparation of the treatise, and in the course thereof,
analyses of conditional propositions of both three and
four terms, in all forms in which they can be expressed,
were made ; and the study of their results led to
the gradual unfolding of the doctrine of Sorites con-
tained in the second of the following chapters.
That doctrine is the culmination of the Science of
Logic, which without it has hitherto been incomplete.
The treatise, up to this point, had been written
wholly in short-hand, and to guard against the possi-
bility that the discovery might be lost if the author
should not live to finish it, and the notes should not be
deciphered, these chapters were written out in full, and
put in position where they would be found and pub-
lished, in such contingency.
But, inasmuch as the work yet remains to be com-
pleted, and the notes to be written out (which can only
be done by the author, his system of short-hand being in
many respects peculiar), its appearance will be consider-
Vi PREFACE.
ably delayed ; and as the discovery, when made known,
will, it is believed, not only be an occasion of interest
from a scientific point of view, but will prove also to
be of practical utility, the author has determined to
publish these two chapters in advance. The chapter
on Enthymemes is published as preliminary, and to ex-
hibit the synopses therein contained (of which the last
shows all valid simple Syllogisms [of three terms] at full
length and in regular form), in connection with those
contained in the chapter on Sorites (Syllogisms of four
terms), thus bringing together, as it were in one view,
all possible valid forms of categorical reasoning. To
those for whose benefit they are thus published the
chapters may seem to be unnecessarily diffuse and
minute, but to condense them would involve very con-
siderable labor, and they are therefore put forth in the
form in which they were written to take their appro-
priate places in the full treatise, trusting that their
minor defects and redundancies may be overlooked.
If the remainder of the treatise shall never appear
from the author's pen, there will be little or nothing
lost. The suggestion herein made, if it have any merit,
will lead other and abler pens to supply the desideratum.
Brooklyn, January 14, 18S8.
UITIVBIISITY]
OF ENTHYMEMES.
§ 1. We have hitherto considered the process of
reasoning with three terms, categorically, in its full
expression, and have examined all the possible forms
of such expression. Such forms are seldom resorted
to, either in common conversation or formal discourse,
whether spoken or written, but abridged forms of argu-
ment are employed in which only part of the process
is expressed, the remainder being implied, and being
usually so obvious as not to require expression. AVe
come now to consider sucli abridged forms.
They are called Enthymemes.
§ 2. An Enthymeme is a Syllogism of which but two
propositions are expressed, the third being implied.
Enthymemes are of three orders ;
1st. That in which the major premise is implied.
2d. That in which the minor premise is implied.
3d. That in which the conclusion is implied.
The following are examples.
Of the first :
N - d;
.-. X - j.
Of the second :
D-j;
•• N - j.
8 LOGIC AS A PURE SCIENCE.
Of the third :
D - ],
and N — d.
In each case the three terms requisite to make up
a full Syllogism appear, and the implied premise or con-
clusion can be readily supplied.
Enthymemes of the first order are herein called
Minor, and those of the second order Major Enthy-
memes, from the names of their expressed premises,
respectively.
§ 3. As every Enthymeme, together with its implied
premise or conclusion, is a Syllogism, it is evident that
only such can be valid as are symbolized by the letters
by which the expressed propositions are symbolized, in
the combinations of vowels symbolizing the propositions
of all allowable moods of categorical syllogisms, as
hereinbefore shown.
By reference thereto, it will be found that all valid
Enthymemes must consist of propositions of which the
following are the symbols ; namely.
Of the first order.
{Minor Enthymemes.)
— , A, A ;
— A, B;
-A, I;
— A, O;
— , E, E;
— E, 0;
-, I, I;
-, I, 0 ;
-, 0, 0.
Of the second order.
{Major Erithymemes.)
A, -A
A,—, B
A,-, I
A, - 0
E, - E
E, - 0
I, -, I
0, -, 0.
Of the third order.
A, A,-
A, B, —
A, I, -
A, 0, -
B, A, -
E. I, —
I, A,-
0, A, -.
ENTHYMEMES. 9
The symbols of minor and major Enthymemes are the
same, except that there is no valid major Enthymeme in
I, O. There are no valid minors in E, O, except useless
ones. Leaving the latter out of consideration, it will be
found that A occurs four times as the symbol of the pre-
mise, and but once as the symbol of the conclusion in
both minor and major Enthymemes ; E once in minors
and twice in majors as the symbol of the premise, and
twice in each as the symbol of the conclusion ; I twice in
minors and once in majors as the symbol of the premise
and twice in each as the symbol of the conclusion ; and
O once as the symbol of the premise and three times as
the symbol of the conclusion in both minors and majors.
Minor Enthymemes are the most common, the sup-
pressed major premise being usually a geneml rule,
readily recognized and acquiesced in without being
expressed.
Enthymemes of the third order are seldom employed,,
except in combination with one of the first or second
order. They will be referred to when we come to the con-
sideration of Sorites, and it will be found that they occur
sometimes in the order of the symbols above shown,
namely, major premise first, and minor second ; and some-
times in the reverse order, minor first, and major second.
§ 4. To the three orders may be added a fourth ; viz.,
an Enthymeme with but one expressed and two implied
propositions. Every demonstrable categorical proposi-
tion, put forth independently as the expression of a
judgment, is such an Enthymeme, being the conclusion
of two implied premises. If the question is asked,
''What is N?" the answer must be either a random
10 LOGIC AS A PURE SCIET^CE.
expression in the form of a proposition, but meaningless,
or the result of thought more or less deliberate, and
therefore based upon some reason, which, as we have
before seen, is a just (or assumed as just) ground of con-
clusion. This ground must be a mental comparison of
the subject, IN", with some other term, and of that again
with the term predicated of the subject in the answer.
The premises thus formed, but not expressed, must be
obvious to the questioner, when the answer is given, and
therefore admitted ; or otherwise explanation would be
demanded. Were this not so, there could be no reason-
ing without going back in every process to some inde-
monstrable proposition (axiom or postulate), or to the
Great First Cause; mth which or with Whom, when
reached in the process of investigation, we must necessa-
rily set out in retracing our steps in the deductive pro-
cess of reasoning.
Such an Enthymeme may also consist, in so far as it is
expressed, of a single proposition put forth as a premise
(usually the major), the unexpressed premise and conclu-
sion being left to be gathered from the attending circum-
stances or from the subject-matter under consideration.
§ 5. The middle term will of course be that term of
the expressed premise, in minor and major Enthy-
memes, which is- not common to both propositions, and
in Enthymemes of the third order, that which is common
to both ; and will vary in position according to the fig-
ure, and the character of the premise, whether minor or
major. In minor and major Enthymemes it may or may
not be distributed, according to the mood, and character of
the premise, whether minor or major ; but in Enthymemes
of the third order must be at least once distributed.
ENTHYMEMES. 11
§ 6. It is manifest, that there are three, and can be but
three, Enthymemes ha\ing two expressed propositions,
viz., one minor, one major, and one of the third order,
in each allowable mood of the syllogism ; and as the
number of such moods is twenty-four, including the use-
less ones, so the number of Enthymemes of each kind is
limited to twenty-four.
The follovv'ing are synopses of all possible valid forms
of categorical Enthymemes of two expressed propositions,
together with the implied premise or conclusion of each,
as the case may be. On the first page of each of the two
synoiDses of minor and major Enthymemes the forms of
the expressed propositions are printed in full, each but
once, in the order A, I, E, O, of the symbols of the con-
clusion, but on the second page they are printed in full
throughout. Where they are repeated, they will be
found to have in each case a different implied proposi-
tion. By counting, it will be found that there are fifteen
forms of the expressed propositions of minor Enthy-
memes (of which four are useless) and twelve of majors.
The capital letters in the names of the moods on each
page of the synopses are the symbols of the proposition
or propositions in the column next adjoining.
The synopsis of Enthymemes of the third order, will
serve also as a synopsis of those of the fourth order, as
first described, by considering the words *' expressed"
and ''implied" as transposed in the headings over the
columns of the propositions.
As arranged on page 17, and read across the page, it
exhibits all possible valid forms of categorical reasoning
with three terms, at full length and in regular form, in
the order of the Moods of the Syllogism.
12
LOGIC AS A PUEE SCIENCE.
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the First Order. (Minors.)
IN THE ORDER A. I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS.
Expressed Propositions.
Implied Proposition.
Moods op
III
-Minor Premise. Conclusion.
Major Premise.
Stllogisms.
A, A.
N- d;
.-. N-j.
V D-j.
bArbara.
A, I.
N- d;
••• n - j.
V D-j.
A, a. i. Ist fig.
D- n;
••• n - j.
V D-j.
dArapti.
tt
//
V d-j.
disamis.
>;
n
••• J-d.
V j-d.
brAmantip.
dimaris.
I, I.
n-d;
.-. n - j.
V D-j.
dAni.
d-n;
••• n - j-
V D-j.
dAtisi.
A,E.
N- d;
•.• D-J.
•.• J-D.
cElarent.
cEsare.
E, E.
?^-D
; .-. N-J.
•.• J-d.
cAmsstres.
&-N
; .-. N-J.
•.• J - d.
cAmenes.
A, 0.
X- d
; .-. n -^ J.
V D-J.
•.• J-D.
E, a, 0. 1st fig.
E, a, 0. 2d fig.
D-n;
II
.-. n -^ J.
V D-J.
•.• d -^ J.
fElapton.
bOkardo.
II
//
•.• J-D.
fEsapo.
E, 0.
N-D
; .-. n -^ J.
•.• J-d.
A, 6. 0. 2d fig.
D-N
; .-. n-^J.
•.• J-d.
A,€,o. 4th fig.
I, 0.
n - d
; .-. n -w J.
V D-J.
fEHo.
II
/;
V J-D.
fEstino.
d-n
; .-. n -w J.
V D-J.
fEriso.
//
//
•.• J-D.
frEsimn.
0, 0.
n-^D
; .-. n — - J.
•.• J-d.
bAroho.
ENTHYMEMES.
13
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the First Order. (Minors.)
IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS.
Moods op
KXPBESSBB PBOFOSITIOHS.
iMPLrCD PBOPOSmON.
Syixogisms.
Minor Premise.
Conclusion,
Major Premise.
harbAr-A.
X- d;
..,
^^ - J.
V D - j.
cdArEnt.
X - d ;
.*.
X-J.
V B-J.
a, A. I.
darll.
X- d ;
n - d ;
•'•
n-j.)
n - J. i
•• D - j.
e. A, 0.
ferlO.
X- d;
n - d ;
.*.
•.• ^ - J.
ce^ArE.
X- d;
.-.
X-J.
•.• J - D.
camEstrEs.
5?-D;
.-,
X-J.
••• J - d.
e, A. 0.
festin 0.
X - d :
n - d ;
,*,
n-^J. 1
n-^J. ("
•.• J - D.
a, E, 0.
barOkO
X-D;
n-^D;
,*,
n-^J.
n-^J. ■
•.• J - d.
darAptl.
D - n ;
.-.
n - J-
•.• D - j.
disAmls.
D- n ;
/.
n - j-
•.• d - j.
datlsl.
d - n :
.-.
^ - j-
V D - j.
feiAptOn.
D- n;
.-.
n-^J.
•.• B-J.
hokArdO.
D- n ;
.'.
n-^J.
V d-^J.
ferlsO.
d - n ;
.*.
n-^J.
•.• ^ - J.
bramAntlp.
D- n;
.*.
^ - j-
•.• J - d.
camEnEs.
a, E. 0.
B-X;
&-X;
' ,
X-J.)
n-^J. S
•.• J - d.
dimArls.
D - n ;
.',
^ - j-
••• .] - d.
fesApO.
D- n;
.'.
n-^J.
V J - D.
f regis On.
d - n ;
•*•
n^J.
•.• J - D.
14
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the Second Order. (Majors.)
IN THE ORDER A, I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS.
'*ii
IH
EXPBESSED Propositions.
Impt.tkt) Proposition.
Moods of
m
Major Premise. Conclusion.
Minor Premise.
Syllogisms.
A, A.
D-j;
.-. N-j.
V K-d.
barbAra.
A, I.
D-i;
.-. n - j.
V K-d.
a, A, L 1st fig.
//
II
•.• 11 - d.
darli.
/'
II
V D-ii.
darAptl.
It
n
V d-n.
datlsi.
J-d;
.-. n - j.
V D-n.
bramAntip.
I, 1.
d-j;
.-. n - j.
•.• D-n.
disAmis.
J-d;
.-. n - j.
••• D-n.
ditnAris.
A, E.
J-d;
.-. N - J.
V 5^-D.
camEstres.
//
'/
•.• D-N.
camEnes.
E, E.
^- J
.-. ?^ - J.
•.• ]sr - d.
cdArent.
J-D
, .-. J^-J.
•.• N - d.
cesAre.
A, 0.
J-d,
tt
.-. n-^J.
II
V n-wD.
a, E, 0. 2d fig.
barOko.
ti
II
V D-N.
a, E. 0. 4th fig..
E, 0.
&-J;
,'. n -^ J.
•.• N - d.
e. A, 0. 1st fig.
II
ri
•.• n - d.
ferlo.
II
It
•.• D - n.
felApton.
II
tt
•.• d-n.
ferlso.
J-D;
.-. n->^J.
V N-d.
e, A, 0. 2d fig.
//
II
V n-d.
festlno.
It
11
•.• D-n.
fesApo.
n
It
•.• d - n.
freslson.
0,0.
d-wJ;
.-. n-^J.
V D-n.
boTcArdo.
ENTHYMEMES.
15
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the Second Order. (Majors.)
IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS.
Moods of
Expressed Propositions.
Implied Proposition.
Syllogisms.
Major Premise.
Conclusion.
Minor Premise.
bArbarA.
i D-J;
...
^^ - j.
...
X-d.
cElarEnt.
fi-- J;
.*.
?^- J.
...
N-d.
A, a, I.
D- j;
.-.
11 - j-
...
]^-d.
dAHZ.
D-J;
.*.
11 - j-
•/
n- d.
E, a, 0.
]^- J;
.-.
11-^ J.
•..
N-d.
fEriO.
B- J;
.-.
11-^ J.
..•
n - d.
cEsarE.
,f-D;
.-.
5^- J.
...
X-d.
cAmestrEs.
J-d;
.-.
2^- J.
...
?^-D.
E, a, 0.
a= -D;
.*.
n^J.
•.•
N- d.
fEstinO.
J-D;
.-.
n-^J.
•/
n-d.
A, €, 0.
J-d;
.'.
n^J.
".'
??-D.
bArokO.
J-d;
.*.
n -^ J.
*.*
n-^D..
dAraptl.
D - j ;
/.
n- j.
...
D-n.
disamls.
d - j :
.-.
n- j.
•/
D-n.
dAtisI.
D - j ;
.-.
n-J.
•.'
d-n.
fElaptOn.
&- J;
/.
n -^ J.
'.*
D-n.
bOkardO.
d-^J;
.*.
n-^J.
*,*
D-n.
fErisO.
&- J;
.-.
n-^J.
'.'
d-n.
brAmantlp.
J-d;
.*.
n- j.
•.•
D-n.
cAmenEs.
J-d;
.-.
^-J.
'.•
D-I^.
A, €, 0.
J-d;
.*.
n-wJ.
".'
B--]^.
dimarls.
j-d;
.*.
n - .]'•
•.*
D-n.
fEsapO.
J-D;
.-.
n-^J.
•.•
D-n.
frEsisOn.
J-D;
•••
n-^J.
•••
d-n.
16
LOGIC AS A PUEE SCIENCE.
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the Third Order.
IN THE ORDER A, I, E, 0, OF THE SYMBOLS OF THE CONCLUSIONS.
)LS OF
E88ED
IITION8.
E2CPRESSED Propositions.
Implied Proposition.
Moods of
^«l
Major Premise. Minor Premise.
Conclusion.
Syllogisms.
A, A.
D — j, and A' — d.
.-. N-j.
harbarA.
A, A.
I) _ j, . X - d.
.-. n - j.
a, a, /. 1st fig.
A, I.
D - j, " n - d.
.-.
daril.
A, A.
D - j, " D - n.
"
daraptl.
A, I.
D - j, " d - n.
.*. II
datul.
I, A.
d _ j, n D - n.
.'. f
disamls.
A, A.
J _ d, " D — n.
.*. "
bramantlp.
I, A.
j _ d, " D - n.
.*. "
dimarls.
E, A.
© _ J, n N - d.
.-. ^-J.
celarEnt.
B, A.
J _ D, " N - d.
"
cesarE.
A, E.
J _ d, " 5^-1).
.'. II
camestrEs.
A, E.
J _ d, " ©• - N.
II
camenEs.
E, A.
© _ J, ,/ N - d.
.-. n-^J.
e, a, 0. 1st fig.
E, I.
B- — J, " n — d.
"
fenO.
E, A.
© _ J, n D - n.
ti
felapiOn.
E, I.
B-_J, n d - n.
"
ferisO.
O, A.
d-^J, " D-n.
, II
bokardO.
A, E.
J _ d, " N - D.
II
a, e, 0. 2d fig.
A, 0.
J - d, " n-_D.
. /'
barokO.
A, E.
J _ d, " B - N.
//
a, €, 0. 4th fig.
E, A.
J _ D, " N - d.
ti
e, a, 0. Sd fig.
E, I.
J _ D, " n — d.
. It
festinO.
E, A.
dF — D, " D — n.
II
fesapO.
E, I.
a= _ B, " d — n.
"
f rests On.
ENTHY3IEMES.
17
Synopsis of all Possible Valid Forms of Categorical Enthymemes
of the Third Order.
IN THE ORDER OF THE MOODS OF CATEGORICAL SYLLOGISMS.
Moods of
Expressed Propositioks. bcpi
UED Proposition,
Syllogisms.
Major Premise. Minor Premise.
CoDclusiou,
hArbAra.
D - j, and X - d.
•. X-j.
cElArent.
3& - J, " X - d.
•. X-J.
A. A. i.
D-j, " X-d.
•• n - j.
dArli.
D - j. " n - d.
•• n - j.
E, A. 0.
& - J. " X-d.
•. n-wj.
jErlo.
B- - J. " n - d.
•. n-«^J.
cEsAre.
^ _ D, " X-d.
•. X-J.
cAmEgtres.
J _ d, " X-D.
•. X-J.
E, A, 0.
^ _ I). r x-d.
•. n-^J.
fE-<fI,-n.
a: _ D. ' n - d.
•. n -^ J.
A. E. c .
J _ d, " X-D.
•. n-^J,
hArOko.
J-d. " n-^D.
•, n-^J.
lArApti.
D-j, '. D-n.
•. u - j.
aisAyni?.
d-j, ■' D - n.
•• n - j.
dAtlsi.
D - j,, n d - n.
•• n - .j-
fElApton.
D-J, '' D - n.
•. n-^J.
bOkArdo.
d -^J. " D — n.
•. n-^J.
rErlso.
D - J, " d - n.
•. n-^J.
brAniAntip.
J _ d,, r. D - n.
•• 1^ - j-
cA?nE>tes.
J _ d, n D - X.
•. 5f-J.
A. E. 0.
J _ d, " D - X.
•. n-^J.
dImAni.
j _ d,, " D — n.
•• n - j.
TEsApo.
J _ D, ' D - n.
•. n -^ J.
rEfilson.
J _ D, " d - n.
•. n-^J.
18 LOGIC AS A PURE SCIENCE.
§ 7. The following will serve as rules by which the
implied proposition of every Enthymeme having two
expressed propositions may be supplied, the first being
applicable to those of either the first or second order,
and the second to those of the third.
1st. The term of the conclusion of an Enthymeme
of either the first or second order which is
common to both expressed propositions deter-
mines the character of the expressed premise,
whether minor or major, according as the
same is either the subject {minor term) or
predicate (major term) of the conclusion, and
the implied premise may be found by com-
paring the other two terms.
2d. The term of the expressed minor premise of
an Enthymeme of the third order not common
to both premises is the subject, and that of
the expressed major premise not common to
both is the predicate, of the implied conclu-
sion, which is universal or particular, and
affirmative or negative, as called for by the
premises.
OF SORITES.
§ 1. We come now to the consideration of reasoning
with four terms, categorically; and we shall herein-
after find that that is the limit beyond which the
human mind cannot go.
§ 2. If we set out to make an investigation concern-
ing any subject, N, and, in the process of our investiga-
tion, become possessed of three judgments, which we
put forth in the form of propositions, as follows :
N — d;
D - j;
J -X,
we may at once apply to such propositions the dictum
of Aristotle, by extending it, as follows —
I first repeat the dictum :
"Whatever is predicated (/. e., affirmed or denied)
universally, of any class of things, may be predicated, in
like manner {viz., affirmed or denied), of any thing com-
prehended in that class."
As extended it will read :
Whatever is predicated (L e., affirmed or denied) uni-
versally, of any class of things, may be predicated, in
like manner (viz., affirmed or denied), of any class com-
prehended in that class; and, in like manner, of any
thing comprehended in any class so comprehended.
20 LOGIC AS A PURE SCIENCE.
We have in our last proposition predicated X (x) of
the whole class J, and in the second proposition have
shown that the class D is comprehended in the class J.
X (x) may therefore be predicated of the class D. But
we have also show^n in the first proposition that [N" (which
may be either a class, or some single thing) is compre-
hended in the class D. We are therefore warranted, by
the dictum as extended, in predicating X (x) of N ; viz. :
X — X.
Stating the propositions in their reverse order, and
appending to them the proposition thus justified, with
the word '^therefore" prefixed, we shall have the follow-
ing expression, which is a Sorites ; viz. :
J
—
x;
D
—
j;
X
—
ci;
.-. X
X.
But we may, without reversing the order of the
propositions, append the new^ proposition, and will have
the same Sorites, but in a different form ; viz. :
X -
- d;
D -
- J;
J -
- x;
X -
- X.
The conclusiveness of the reasoning in both forms is
apparent.
§ 3. Thus we have a complete Syllogism (but in two
different forms or figures) consisting of four propositions,
composed of four terms.
SORITES. 21
Let lis now proceed to analyze it, and in the course
of the analysis I shall give new names to the terms and
propositions, which mil be used when referring to them
as parts of the Sorites, so as to distinguish them from
like parts of a simple Syllogism, which will be called,
when referred to as such, by their old names.
And 1st, as to the terms.
The subject, X, with which we set out, is equiva-
lent to the minor term as we have hitherto employed
it. I call it the magnus term of the Sorites, in the
sense of holding a chief position ; it being the principal
thing about which we are concerned.
The two terms, D and J, are each greater {major)
than the magnus term in the forms above exhibited
(which you ^vill hereinafter lind are the perfect forms),
but one, D, is less {minor) than the other, J. They are
both middle terms, and are each once distributed, and
are compared, one with one of the other terms, and the
other with the other, in the first and third propositions,
and with each other in the second. They will be called,
D, the minor -middle^ and J, the major-middle terms.
The term X is equivalent to the major term as
hitherto employed, but is greater than the major-middle
term, and is the greatest of all the terms of the Sorites.
It will therefore be called the maximus term.
The four terms, as in the case of a simple Syllogism,
occur twice each, the magnus and maximus terms each
once in the premises (first three propositions) and once in
the concluding proposition, and the minor-middle and
major-middle terms each twice in the premises.
22 liOGIC AS A PURE SCIENCE.
N and X are letters in the words jnagnus and maxi-
TTius respectively, and will serve to keep their logical
significations in mind, in like manner as the letters N,
D, and J, in the words minor ^ Tniddle, and Tnajor^ have
hitherto served in respect to their logical significations :
but they will not in their future use so serve invari-
ably.
2d. As to the propositions.
Three are premises, and will be called from the names
of the terms occurring in them respectively :
The magnus premise ;
The middle premise (omitting the prefixes minor and
major as unnecessary, there being no middle premise
in a simple Syllogism) ;
The maximus premise.
The concluding proposition will hereinafter be found
to be the ultimate one of two conclusions warranted by
the premises ; and to distinguish it as such, I shall call
it the ultima (conclusio understood).
8d. As to the figure.
The figure of a simple Syllogism depends upon the
positions of its terms, but that of a Sorites upon the
positions of its magnus and maximus premises. It
will be called the configuration. There are two, the first
called regressive^ in which the maximus premise is the
first, and the magnus last ; and the second, progressive,
in Avhich the magnus premise is the first and the maxi-
mus last. The progressive configuration was the only
one known until about the beginning of the seventeenth
SORITES. 23
century, when the regressive was discovered by a Ger-
man logician named Goclenius ; and it is called also
Goclenian after him. It has been a subject of dispute
among logicians as to which configuration should be
called progressive, and which regressive, but the prevail-
ing opinion is in favor of the names as herein used.
They are generally treated of in the order as in the last
sentence ; but I have reversed it, exhibiting the regress-
ive first, and the progressive last. The moods of each
configuration, and their number, will hereinafter appear.
§ 4. If all Sorites, in respect to the positions of the
teims, were in the forms hereinbefore given, and their
conclusiveness were equally as apparent, I might at once
proceed further to illustrate and comment upon them,
and state the rules usually given in logical treatises con-
cerning them, which are applicable only in such case ;
but such is not the case, and I defer further comment
until I jshall have exhibited them in another aspect in
which they can be considered ; viz., as complex expres-
sions consisting of two Enthymemes.
The Sorites, so to be exhibited, will be the same as
before given ; and for the sake of brevity, I shall call
the terms and propositions by the names hereinbefore
given to them, in advance of exhibiting them under the
new aspect.
§ 5. For the purpose of such consideration I repeat
the three propositions with which we set out.
X — d;
D- j;
J — X.
24 LOGIC AS A PURE SCIENCE.
If now, having possessed ourselves of these judgments,
but failing to observe, from their perfect concatenation,
that we may at once deduce from them the ultimate con-
clusion wrapped up in them, we proceed to syllogize
with them by means of simple Syllogisms of three prop-
ositions, we shall naturally commence with the widest
truth which we have discovered, viz., J — x ; and we
shall find our first Syllogism to be as follows :
J — x;
D- j;
.-. D - X,
and, having thus become possessed of a new truth, viz.^
D — X, we shall put it forth as a premise, combining
with it our first proposition, as yet unemployed, and
produce a second Syllogism as follows :
D
—
x;
N
—
d;
N
X.
The conclusion of this second Syllogism is the ultima
of the Sorites, as we have before seen it.
But if, in the course of our investigation, we had
stopped after the discovery of the first two truths, viz. :
K — d;
D - J,
and had syllogized with them, we should in like manner
SORfTES. 25
have commenced with the widest truth then discovered,
viz., D — j, and our first Syllogism would have been :
D - J:
N — d;
.•• ^ - j.
The question would then naturally have arisen, But
what is J? and resuming the process of investigation,
we should have discovered that J — x, and thereupon
would have syllogized again :
J -x;
^'- J;
.*. N — X,
and thus, by a second series of Syllogisms, we should
have arrived at the ultima of the Sorites, as we have
before seen it.
By the former process, we retraced our steps after
having reached the summit of our investigation, and
it is therefore properly called regressive ; by the latter
w^e have reasoned as we progressed, and it is therefore
properly called progressive ; but by both processes we
have arrived at the same ultimate conclusion, illustrat-
ing the axjhorism that ''all truth is one."
The middle premise, as you will observe, is the minor
premise of the first Syllogism in the first series, and the
maxim us premise the major ; and the middle premise
is the major premise of the first Syllogism in the second
series, and the magnus premise the minor ; and all the
26 LOGIC AS A PURE SCIENCE.
Syllogisms are in Barbara in the first figure, which you
have learned is the only j)erfect figure.
§ 6. But we may reason imperfectly, and that too,
even when we have our judgments in a perfect concate-
nation, as they have thus far been exhibited ; and, in
such case, we shall find our Syllogisms to be in one or
more of the imperfect figures. If, in the regressive
process we begin to syllogize with the middle premise
as the major premise of the first Syllogism (instead of
the minor), and the maximus as the minor (instead of
the major) ; and in the progressive process, wdth the
middle premise as the minor premise of the first Syllo-
gism (instead of the major), and the magnus premise as
the major (instead of the minor), we can frame, or
attempt to frame, two other series of Syllogisms, which I
here exhibit, with the two Syllogisms of each series,
side by side, as follows :
In fhe regressive process.
D — j; X — d;
J — x; y^ N — d;
.-. X — d. — -^
In the progressive process.
N — d; J — x;
D — j ; ^ J — 1^ '
.-. j — n. — —^^^^ .*. 11 — X.
In the latter series, only a particular ultimate con-
clusion is arrived at ; in the former, no ultimate conclu-
sion is warranted by reason of non-distribution of the
middle term in the second attempted Syllogism.
SORITES. 27
Thus, as you will perceive, imperfect processes are
followed by imperfect or no results.
§ 7. To recur now to the two principal series, and for
the purpose of bringing the two Syllogisms of each
together, in such a method of arrangement that you may
at once see the connection between them, and the appli-
cation of the remarks that are to follow, I repeat them,
putting the two Syllogisms of each, side by side.
First, or regressive series.
J — x; ^^ ^D — x;
D-j; / N-d;
.-. D —
Second, or progressive series.
D-j; J - x;
X-d; ^ N-j;
.-. X - j. -^ .-. X - X.
By taking an Enthymeme of the third order from the
first Syllogism, and one of the first order from the second
Syllogism of the first series, and putting them together
in one expression, and, by taking an Enthymeme of
the third order from the first Syllogism of the second
series, hut transposing the propositions so taken^ and
one of the second order from the second Syllogism of
the same series, and putting them together in one ex-
pression, we shall have the same Sorites, as before, in
the two configurations, viz.:
28 LOGIC AS A PURE SCIENCE.
Regressive Sorites Proyressive Sorites
from the first series. from the second series.
J — x; X — d;
D- j; D - j;
X — d; J — x;
.-. X — X. .-. N — X.
The conclusion of the first Syllogism in each series is
held in the mind (otherwise there were no Enthymeme),
but carried forward mentally, and employed as a pre-
mise, still unexpressed, in connection with the P]nthy-
meme taken from the second.
A Sorites considered as a complex expression as
above shown is also called a Chain-Syllogism.
§ 8. The middle premise (being the proposition B — j
in which the minor-middle and major-middle terms are
compared) will always be the second proposition in every
Sorites, simple (as hitherto shown) or compound (as to
which latter you will hereinafter be instructed) ; and by
expressing it, in connection with the ultima, every
Sorites may be still further abridged, thus :
D- j;
.-. N — X.
All the four terms here appear, but each only once.
Such an expression is in the form of an Entliymeme
(but is not an Enthymeme, for that can have only three
terms), and may properly be called an Abridged Sorites.
From the employment of the middle premise as the
minor or major premise of the first Syllogism, I desig-
nate Sorites (considered as complex expressions) minor
SORITES.
29
and major Sorites, respectively, for the purpose of classi-
fication as hereinafter shown. Either may be regressive
or progressive ; but we shall see that the proper division
of Sorites is into regressives and progressives.
Observe, that in all major Sorites, but in no minors,
the premises constituting the Enihymeme of the third
order taken from the first Syllogism, are transposed.
§ 9. The Syllogisms of the two principal series (of
Enthymeines of which the Sorites exhibited consist)
are Avholly in the first figure. But a little reflection
will show that Sorites may also consist of Enthymemes
taken from Syllogisms in any of the figures capable of
combination in series, quantity and quality considered.
And, as all Sorites may be abridged in the manner
hereinbefore shown, it is also manifest that the range of
possible abridged Sorites is limited to the number of
possible combinations of two proj)ositions composed of
four terms, expressed in the same form as to the order
of the temis throughout, but modified in respect to
quantity and quality, as in the following scheme ; and
only such can be valid as are capable of being expanded
into full Sorites, and from full Sorites into at least two
series of Syllogisms. The propositions must be in one or
another of the combinations shown by full lines in the
scheme.
D — i ; -^:::i ^rr-^^ X — x.
X.
11 — X.
X.
30 LOGIC AS A PURE SCIENCE.
Considering the lines connecting the propositions,
each as signifying ''and therefore," there are sixteen dif-
ferent combinations. But of these, only nine will be found
to be valid, and they are symbolized by the same symbols
as those of valid Enthymemes of the first order, as here-
inbefore shown, and may be expanded into full Sorites
(the supplied premises varying in the order of the terms
as well as in quantity and quality), and from full Sorites
into two, three, or four series of Syllogisms, with the
middle premise as either the minor or the major premise
of the first Syllogism of one or more series, excei^t in two
cases, which will be hereinafter noted.
The number of valid full Sorites into which the nine
abridged forms may be so expanded is one hundred and
forty-four, of which one half are minors and one half
majors, classified as such according to the combinations
of the symbols of the abridged forms, as follows :
Symbols.
Mi7iors.
Majors,
A, A.
1
1
A, E.
4
8
A, I.
16
10
A, 0.
24
24
E, E,
4
4
E, 0.
10
16
T, I.
4
4
I, 0.
8
4
0, 0.
1
1
72 72
The following synopsis exhibits all possible valid cate-
gorical Sorites, in their abridged forms, as minors on the
SORITES. 31
left-hand pages, and as majors on the right ; together with
the premises by which they may be expanded into valid
full Sorites, and the names of the moods in which they
can be further and fully expanded into series of Syllo-
gisms. They are arranged in the order A, I, E, O, of
the symbols of the ultima.
The abridged forms may be expanded into full Sorites
by writing first, the first of the two supplied premises ;
secondly, the middle premise; thirdly, the second of the
two supplied premises ; and lastly, the ultima.
Preceding the synopsis are given two series of schemes,
by which the different ways in which abridged Sorites
may be expanded into full Sorites, and from full Sorites
into series of Syllogisms, in all combinations of figures
in which they are capable of being so expanded, may be
clearly seen. The terms of the abridged Sorites are in
capitals enclosed in circles connected by lines represent-
ing the copulas of the propositions. The curved lines
(considered as copulas) above the propositions constitut-
ing the abridged Sorites, in connection A\ith those propo-
sitions, indicate two expanded Sorites, and in connection
also with the dotted straight line above, indicate two
series of Syllogisms ; and the lines below, two other
expanded Sorites, and two other series of Syllogisms.
Tlie dotted straight lines show the unexj)ressed conclu-
sions of the first Syllogisms, which in each case becomes
one of the premises of the second. The modifications of
the propositions of the abridged Sorites, in respect to
quantity and quality, are indicated by the symbols above
and below the lines representing their copulas respect-
ively ; those above referring to the Sorites and Syllo-
32 LOGIC AS A PUEE SCIENCE.
gisms indicated above, and those below, to those below.
The modifications of the other indicated propositions
are also in like manner signified.
The symbols in connection with the lines are those
only in which the Sorites and Syllogisms are valid in the
figures shown.
It is not meant that each symbol in connection with
each other will yield a valid Sorites, but that each, in
<;onnection with some one or more of the others, will
be found valid. Thus, in the second scheme of minors,
the maximus premise, A, will combine with the middle
premise as E or O, and E with A or I, but not other-
wise.
The designations of premises, written between paral-
lel curved lines, refer to the propositions indicated by
both lines ; the symbols and number of the figure being
on the other side of each line, respectively.
By marking all the lines with all the symbols, you
will be able to make an exhaustive analysis of all possi-
ble ways in which it may be attempted to frame simple
Sorites. In view of the number given on the next page,
you may think the attempt formidable, but you will find
it not so much so as it will at first appear, if you but
consider and apply to the symbols the rules of the syllo-
gism before proceeding to test them. The lines above the
proi3ositions constituting the abridged Sorites are marked
with all the symbols of the propositions respectively, as
they may be employed in single simple syllogisms, as
hereinbefore shown, but those below, not ; and if you
first add to the latter the omitted symbols, making them
to correspond with those above, you will find that such
SORITES. 33
added symbols will, in all cases, yield no conclusion in
the second of the Syllogisms, by reason of one or the
other of the two faults, undistributed middle and illicit
process of the major. If the remaining symbols be then
added to each line, a violation of some one or more of
the rules of the syllogism will be found in either the
first or second Syllogism.
The total number of the ways in which it may thus
be attempted to combine the four symbols A, E, I, O.
according to the schemes is eight thousand one hundred
and ninety-two, that being the product of the number of
ways (256) in which the four symbols may be combined
(all the same, or partly the same, or all different), multi-
plied by the number of combinations of propositions (4)
indicated by each scheme, and again by the number of
schemes (8)— (256 x 4 x 8 = 8192). .
The total number of valid Sorites without regard to
their character as minor or major, or as regressive or pro-
gressive, will be hereinafter found to be forty-four.
By examining each scheme, and comparing the
Sorites and series of Syllogisms thereby indicated (those
above with each other, and those below with each other),
and by comparing each scheme with each of the others
in all possible ways, the differences between, and con-e-
lations of, the several figures of the Syllogism and
the two kinds of Sorites indicated by the schemes (that
is, either minor or major), will also clearly appear, and
the student cannot fail to be impressed with the har-
mony and symmetry of pure reasoning, in all its varied
possible forms of expression.
34
LOGIC AS A PURE SCIENCE.
SCHEMES OF MINOR SORITES.
FIRST SYLLOGISM IN FIRST FIGURE.
A J^E . or O. ^ :^
1 __- isUig,^ ^^^^iPren^ .
4. J p, A. or E. _J^^^\V^
^^. I. or E.
FIRST SYLLOGISM IN SECOND FIGURE.
]5-_or_(). i.
@4^M^0 •■• 0-1^©
. ^ A. orE. StA3^^'"^VN^
^ -f^> __^-— ^^^^
SORITES.
35
SCHEMES OF MAJOR SORITES.
FIRST SYLLOGISM IN FIRST FIGURE.
A J:-5-_"J-_0:
Min. prerti. A^ o^ E.
1st
A. or I
A.L or £.
FIRST SYLLOGISM IN SECOND FIGURE.
E. or O.
Jst fig.
E.
LOGIC AS A PURE SCIE]S^CE.
SCHEMES OF MINOK SORITES.
FIRST SYLLOGISM IN THIRD FIGURE.
I-P£9-_
FIRST SYLLOGISM IN FOURTH FIGURE.
I. E. or O.
o^T
A.o^
I. or E.
SORITES.
37
SCHEMES OF MAJOR SORITES.
FIRST SYLLOGISM IN THIRD FIGURE.
f
^ J'_orJl-
Min. prem.
^^^^. ^l: -^^^ ^ret^^-
A. or I.
i:
FIRST SYLLOGISM IN FOURTH FIGURE.
^ OP THE "nP^^
1.^-0, ffUH I V E E SI T 1
r ^!^^
-^^ orT - — ^*^=^ — i
A. or E.
______
1
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms of
TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS
PREMISES,
Sym-
BOLS.
Nos.
Abridged Minor
Sorites.
Middle Premise, Ultima.
Mood of
First
Syllogism.
Major Premise
OF First
Syllogism.
Minor or
Major Premise
of Second
Syllogism.
Mood of
Second
Syllogism.
A, A.
1
D-j; /. N-x.
bArbAra
J-x
N-d
barbArA
A, I.
2
D— j ; .*. n — X.
bArbAra
J-x
N-d
a,AJ. 1st fig.
3
// /'
>i
II
n-d
darll
4
// It
\ or, A, A, I ■
ri
D-n
J darAptI
\ disAmls
5
rt If
bArbAra
II
d-n
datlsl
6
II II
brAmAntip
N-d
J-x
dAtisI
7
II If
dlmAris
n-d
II
It
8
II II
dArApti
D-n
II
It
9
II II
dIsAmis
d-n
If
It
10
II II
J bArbAra 1
1 or, A,A,i \
J-n
D-x
J dAraptI
1 dAtisI
11
II n
bArbAra
1'
d-x
dlsamls
12
II It
II
If
X-d
brAmarUIp
13
II ti
II
II
x-d
dimarls
14
II II
dArApti
D-x
J-n
disAmIs
15
If If
dIsAmis
d-x
//
It
16
11 If
brAmAntip
X-d
//
II
17
II If
dImAris
x-d
//
II
1,1.
18
d-j; .-. n-x.
dArIi
J-x
D-n
disAmIs
19
fi II
dAtlsi
D-n
J-x
dAtisI
20
If It
dArn
J-n
D-x
If
21
II II
dAtlsi
D-x
J— n
disAmIs
SORITES.
39
Abridged Categorical Sorites.
AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY
EXPANDED.
Sym-
bols.
N08.
Abridged Major
Sorites.
Middle Premise. Ultima.
Mood of
First
Stllogism.
Major or
Minor Premise
of Seookd
Syllogism.
Mood of
Second
Syllogism.
1
A, A.
D-j; .-. N-x.
bArbAra
N-d
J-X
/jArbarA
A, I.
2
D — j ; /. n — X.
j bArbAra 1
1or, ^,^,1 f
N'-d
J-X
SA.a.I
} dAril
3
tt It
dArIi
n-d
"
f
^
ir It
dAvApti
D-n
It
5
n It
dAiM
d-n
ft
It
6
tt II
brAmAntip
J-x
D-n
dimArls
7
ri II
dArApti
D-x
J-n
»
8
ft II
dAtM
d-x
tr
II
9
ti II
J bArbAra I
"1 or, A,A,i S
X-d
If
j braiPAntIp
1 dimArls
10
It II
dArIi
x-d
It
11
11
tt II
brAmAntip
J-n
D-X
dAHI
1,1.
12
d— j ; .*. n— X.
dIsAmis
D^n
J-x
dAril
13
// It
dImAris
J-x
D-n
dimArls
14
II It
dIsAmis
D-x
J-n
It
15
It tt
dImAris
J-n
D-x
dAHI
40
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms of
TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS
PREMISES,
Sym-
bols.
Nos.
Abridged Minor
Sorites.
Middle Premise, Ultima.
Mood of
First
Syllogism.
10
III
Minor or
Major Premise
OF Second
Syllogism.
Mood of
Second
Syllogism.
A, E.
22
D-j ; .-. S-X.
cElArent
^-X
N-d
celArEnt
23
// n
cEsAre
X-J
//
"
24
ti 11
cElArent
d^-N
X-d
cAmenEa
25
n II
cEsAre
}^-J
II
II
E, E.
26
B-_J; ... 5?_x.
cAmEstres
x-j
N-d
cdArEnt
27
II II
cAmEnes
N-d
x-j
cAmenEs
28
II II
II
X-d
N-j
CdArEnt
29
II II
cAmEstres
N-J
x-d
cAmenEs
A, 0.
30
D-j; .-. n-^X.
cElArent
dF-X
N-d
j e, A, 0.
\ Ist fig.
31
// n
II
II
n-d
ferlO
32
II II
II
■ or, E, A,o\
1'
D — n
SfelAptOn
\ hoklrdO
33
n II
cElArent
II
d-n
ferlsO
34
It II
cEsAre
X-J
X-d
J e, A, 0.
\ l.st fig.
35
II II
n
II
n-d
ferlO
36
II II
\ or, E, A,o]
II
D-n
jfef,Aj)tOn
1 bokArdO
37
II It
cEsAre
II
d-n
ferlsO
38
II II
cElArent
dF-N
X-d
j A, e, 0.
\ 4rh fig.
39
II II
cEsAre
5?-J
"
//
SORITES.
41
Abridged Categorical Sorites. (Continued.)
AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY
EXPANDED.
Sym-
bols.
Nos.
Abridged Major
Sorites.
Middle Premise. Ultima,
Mood op
First
S-XXLOGIBM.
1 .
Ill
PI
Major or
Minor Premise
OP Second
Stllogism.
Mood of
Second
Syllogism.
A,E.
16
D-j;
.-. J^-X.
bArbAra
N-d
J_X
cElarEnt
17
n
ft
"
It
X — J ! cEearE
18
It
II
"
X-d
J — X camEnEs
19
ti
It
II
It
N-J
camEstrEa
20
It
It
cAmEnea
J-X
N-d
cesArE
21
ft
tt
cAmEstres
X-J
II
It
22
It
tt
cAmEnes
J-N
X-d
cAmestrEg .
23
II
II
cAmEstres
}?-J
//
II
E, E.
24
B-J;
.-. ^-X.
cElArent
N-d
X-j
cAmestrEs
25
II
II
cEsAre
X-j
N-d
cesArE
26
It
If
n
N-j
X-d
cAmestrEg
27
II
It
cElArent
X-d
N-j
cesArE
A, 0.
28
D-j;
.-. n-.^X.
S bArbAra \
1 or, A,A,i\
N-d
J-X
\f^S
29
'/
;/
dArR
n-d
If
It
30
/;
//
dAvApti
D-n
"
ft
31
II
tt
dAtlsi
d-n
II
tl
32
II
It
^ bArbAra 1
1 or, A, A, is
N-d
X-J
\Ea. 0
IfEstinO
33
It
It
dAvR
n-d
"
tt
34
II
II
dArApti
D-n
//
It
35
tt
II
dAtM
d-n
//
It
36
It
It
bArbAra
X-d
J^-N
J a, E, 0.
\ 4th fig.
37
II
tt
It
It
N-J
j o, E, 0.
1 2d fig.
38
tl
tt
It
II
n-«-,j' barOkO
1
42
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms of
TOGETHER WITH ALL THEIR POSSIBLE MAGNUS
PREMISES,
AND MAXIMUS
1
a
1 w
Sym-
bols.
N08.
Abridged Minor
Sorites.
Middle Premise. Ultima.
Mood of
First
Syllogism.
Minor or
Major Premie
of Second
Syllogism.
Mood of
Second
Syllogism.
A, 0.
40
41
I)_j; ... ll__X.
n tt
fElAptm
hOkArdo
B^-X
d-^X
J-n
bokArdO
II
42
It It
fEsApo
5-D
It
II
43
It It
\hArhAra \
1 or, A,AA\
J-n
D-X
SfElapt On
TfErisO
44
II It
hArhAra
//
d-wX
bOkardO
45
II II
II
\ or, A, A,i
It
X-D
SfEsapO
ifrEsisOn
46
II II
hrAniAntip
N-d
J-X
fErisO
47
II r
dlniAris
n-d
ti
11
48
It 1'
dArApti
D~n
II
49
11 r
dIsAmis
d-n
1'
"
50
II 1'
brAmAntip
K-d
X-J
frEsisOn
51
II n
dImAris
n-d
/'
II
52
II It
dArApti
D-n
I'
II
53
It "
dIsAmis
d-n
II
I, 0.
54
d-j; .-. 11 ^X.
/Brio
^-X
D-n
bokArdO
55
II It
fEstIno
X-J' "
It
56
II It
fErlso
^-X\ J-n
"
57
'1 II
frEsIson
X-D
'/
It
58
II II
dAtlsi
D-n
^-X
fErisO
59
It It
II
It
X-J
frEmOn
60
n II
dArIi
J-n
D-X
fErisO
61
II II
II
X-D
frEsisOn
SORrtES.
43
Abridged Categorical Sorites. (Continued.)
AND
MOODS OF SIMPLE SYLLOGISMS IX WHICH THEY CAN
BE FULLY
EXPANDED.
Sym-
BOLS.
Nos.
30
Abridged Major
Sorites.
Middle Premise. Ultima.
Mood of
First
Syllogism.
Minor Premise
OF First
Syllogism.
Major on
Minor Premise
OF Second
Syllogism.
Mood of
Second
Syllogism.
A, 0.
I)_j: ... n-^X.
IrAmAntip
J — 11
B-X
fEriO
40
'/ It
II
11
X-l)
fmtinO
41
II II
cAmEnes
^-X
■vr .1 j 6, A, 0.
^—^^\ 2dfig.
42
II II
'■
'■
n — d festlnO
43
It II
•'
"
D — n fesApO
44
It f
II
d — n fresh On
45
ir It
cAmEstref!
X-J
if-cij''^-?i«g.
46
It If
It
n — d festInO
47
It 1'
"
II
D — 11 fesApO
48
II II
II
II
d-n
fresIsOn
49
II t'
icAmEnes \
j or, A,E,os
^-N
X-d
SA,e,0
UArokO
50
II '1
J cAmEstres \
\ or, A,E,oS
?^-J
ti
\A,e,0
} bArokO
51
II II
bArOko
n^J
II
It
I. 0.
52
d — j; .-. n-wX. disAmis
D-n
J-X
fEriO
53
i "
/'
X-J
fEstinO
54
II n
dImAris
J-ii
&-X
fEnO
55
It It
It
''
X-D
fEsHnO
44
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms of
TOGETHER WITH ALL THEIR POSSIBLE MAGNUS AND MAXIMUS
PREMISES,
Sym-
Nos.
E, 0.
0, 0.
62
63
64
65
66
67
68
69
70
71
Abkidged Minor
Sorites.
Middle Premise. Ultima.
^- J ; .-. ii-wX.
Mood op
First
Syllogism.
« £ £
Kg 3
cAmEstres
or, A, E, o
cAmEstres
cAmEnes
x-j
X-d
I or, A,E,o I I
cAmEnes "
N-J
72 d-^J ; .'. n — 'X. bArOko
x-j
§fi«g
X-d
n-d
D-n
d-n
N-j
J-n
X-j
X-d
Mood op
Second
Syllogism.
e,A, 0. 1st fig.
ferlO
SfdAptOn
I boklrdO
ferlsO
^,^,aistfijr.
ferlO
SfdAptOn
1 bokArdO
ferlsO
A,e,0. ith&g.
^,f,^.4tlil
D — n I bokArdO
SOKITES.
45
Abridged Categorical Sorites. (Concluded.)
AND MOODS OF SIMPLE SYLLOGISMS IN WHICH THEY CAN BE FULLY
EXPANDED.
Sym-
bols.
N08.
56
Abridged Major
Sorites.
Middle Premise. Ultima.
Mood of
First
Syllogisic
H
mi
Mood of
Second
Syllogisji.
E, 0.
^_J;... n-^X.
1 or, E,A,o\
N-d
x-J
SA,€,0
bArokO
57
'r /'
fErlo
n-d
''
It
58
f n
fETApton
D-n
II
II
59
" II
fErlso
d-n
"
It
60
tt It
j cEsAre \
\ or, E,A,ofi
^-]
x-d
iA,e,0
\ hArokO
61
n II
fEslIno
n-j
II
It
62
II II
fEsApo
J-n
"
11
63
It It
frEsIaon
j-n
II
"
64
'1 II
cEsAre
X-j
N — d ^,4,0.2dfig.
6b
'1 II
II
It
n-d
featlnO
66
n II
II
II
D-n
fesApO
67
II 1'
If
It
d-n
fresh On
68
It II
cElArerU
X-d
N-j
e.A.aadfig.
69
II II
II
It
n-j
fegtInO
70
II II
II
II
J-n
fesApO
71
If II
It
It
j-n
f res Is On
0, 0.|72
d — ^J ; .*. n — 'X. hOkArdo
D-n
X-j
bArokO
46
LOGIC AS A PURE SCIENCE.
§ 10. By examining the foregoing synopsis and testing
the same, it will be found that
If the major-middle term (predicate of the middle premise) he
the middle term of the first Syllogism, the7i if the Sorites he
Minor :
hut if it be
and the configurations of
the Sorites, whether re-
gressive or progressive, or
bofn, and the number of
each, will be as follows :
Reg., G
Reg.,i 6
Reg., 6
3 I Reg.,' 6
Prog.,
1
6
2
Prog.,
6
2
-4
Prog.,
4
J,
Major ;
and the configurations of
the Sorites, xchethe)- re-
aressive or progressive, or
both, and the number of
each, will be as follows :
2 Reg.,
4 Reg.,
2 Reg.,
Jf I Reg.,
0 Prog., G
4;
Prog.,
3 Prog.,
4
But if the minor-middle term (suhject of the middle pj^emise)
he the middle term of the first Syllogism, then, if as sec-
ondly ahove, the figures of the Syllogisms may he, and the
configurations of the Sorites and the numher of each ivill he
as follotvs :
Minor ;
Major ;
1
1
Prog.,
6
3
3
Reg.,
G
Prog.,
G
1
2
Reg.,
6
Prog.,
6
3
-^
Prog.,
3
1
J^
Reg.,
G
4
1
Reg.,
3
3
1
Prog.,
6
Jf
r,
Reg.,
6
Prog.,
4
3
2
Prog.,
G
Jf
u
39
Prog.,
4
33
3
Jf
Reg.,
3
32
40
Total minors, 72 ; Total majors, 72 ;
Grand total, 144.
SORITES. 47
But, by a careful examination of the synopsis, it will
be found that fifty-six of the Sorites are both minors
and majors. That number must therefore be deducted
from the grand total, leaving eighty-eight different
forms.
Each of the four figures occurs as the figure of the
first Syllogism in both minor and major Sorites ; but the
second does not occur as the figure of the second Syllo-
gism in minors, nor the third in majors. With these
exceptions, all the figures occur also as figures of the
second Syllogism.
The following is a synopsis of all the eighty-eight
possible forms of valid simple Sorites arranged according
to their configurations, regressives on the left-hand pages,
and progressives on the right, and without regard to their
being either minor or major, but showing in the columns
on the left-hand side of each page, the moods of the
Syllogisms in respect to which they are minors, and on
the right, those in respect to which they are majors.
There will be found on the pages of regressives, seven-
teen, and on the pages of progressives, fifteen, in Avhich
the moods are only on one side, leaving twenty-seven
regressives and twenty-nine progressives in which the
moods are on both sides, and which together make the
fifty-six alike on both sides of the preceding synopsis,
as above stated. Two, namely, Xos. 25 and 38, are the
exceptions hereinbefore referred to. No. 25 is a minor
Sorites only, and No. 38 a major Sorites only, in both
configurations.
As before, they are arranged in the order A, I, E, O
of the symbols of the ultima.
48
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms
Series of Syllogisms
IN WHICH THE MIDDLE
If
Regressive Configuration.
Series of Syllogisms
in which the middle
Premise
IS Minor,
Maximus
Major of
o
1
1
Premise is Major,
AND THE
Premise
THE riBST
Maximus Middle
Premise. Premise.
Magnus
Premise.
1
Ultima.
AND THE Maximus
Premise Minor of
THE FIRST.
bArbAra
barbArA
J-X
D-j
X-d
/.X"— x
bArbAra
,a,A,I. 1
1 1st fig. r
2
J_x|D-j
X— d
.-. n — x
1
bArbAra
darll
3
J-x,D-j
n-d
.-. n— X
m
bArbAra
or, A, A, i
davAptl \
disAmIs j
4
J-x D-j
D-n
.-. n— X
(2) (1)
brAmAntip
(3) (4) "
dimArls
bArbAra
datlsl
5
j
J-x
D-j
d— n
.-. n— X
dArIi
disAmIs
1
•^1
J-x
f^-j
D-n
.-. n — X
dImAris
dimArls
dArApti
disAmls
1
D-x
D-j
J-n
.-. n — X
dArApti
dimArls
dAtlsi
disAmla
8
D-x
a-j
J-n
.-. n — X
dIsAmis
dimArls
dIsAmis
disAmIs
9
d-x
D-j
J-n
.-. n — X
dAtlsi
dimArls
brAmAntip
disAmls
10
X-d
D-j
J-n
.*. n — X
I bArbAra
1 or, A,A,i
bramAntIp
dimArls
dim Aril
disAmIs
11
x-d
D-j
J-n
.'. n — X
dArR
dimArls
cElArenf.
celArEnt
12
J-X
D-j
X-d
.-.N-X
cAmEnes
cesArE
cEsAre
cdArEnt
13
X-J
D-j
X-d
.-.X-X
cAmEstres
cesArE
14
x-d
D-j
N-J
.-.X-X
bArbAra
camEstrEs
15
x-d
D-j
J-X
.-.X-X
bArbAra
camEnEs
cAmEnes
celArEnt
16
X— d B--J
N-j
/.X-X
cElArmt
cesArE
cAmEstrea
celArEnt
17
X-j &-J
X^-d
.-.X-X
cEsAre
cesArE
SORITES.
49
of Simple Categorical Sorites.
Series of Stixogisms j
IS WHICH THE Middle '
i
b
O
1
1
..^ ^ Series of Stllogisms
PROGRESSn-E COXFIGUBATION. ^^, ^^.^,^^ ^^^ ^^^^^
Premise
Is Mesor, '
Magnts
Major of
Premise is M ajob,
AND THE
Premise
the first
Magnus
Premise.
Middle '■
Premise,
Maximus
Premise,
Ultima.
AND THE Magnus
Premise Minor op
TUB FIRST.
N-d
1^-J
J-X
.-.X-x
(2) (1)
bArbAra
(3) (4)
bArbarA
hrAmAntip
dAtisI
2
^X-d
T^-J
J-X
. ,, V ^bArbAra
■' ^^ — ^ , or. A, A,i
A, a. I
dAril
film Arts
dAmi
3
n-d
I^-J
J-X
.'. n — X ,dArll
dAril
dArApti
dAtUI
1
4
1
Di^n
I> — J
J-X
.'. 11 — X : dArApfi
\
dAril
dhAmis
dAtisI
5
d-n
1^-J
J-X
.-. n — X dAtisi
dAril
dAtM
dAthI
6
D-n
<^-J
J-X
••• "*-^
dIsAmis
dAnI
hArbAra
or, A, A, i
dAraptl)
dAtlsI S
7
J-n
r>-j
P-X
.-. n — X
brAmAntip
dAril
dArIi
dAthI
8
J-n
d— j
D-X
.*. n — X ,
dlniAris
dAril
hArbAra
dlsa?nls
9
J-n
i>-j
d-X
.*. n — X
bArbAra
brAmantIp
10
J-n
T>— j
X — d
.-. n — X
bArbAra
dimarls
11
J-n
T^-J
x-d
...„-.;
!l2
1
X-d
l>-j
J-X
.-.X-X
bArbAra
cElarErU
1
|13
X-d
D-J
X-J
.-.X-X
bArbAra
cEgarE
cEsAre
cAmenEg
1
|14
^-J
D-J
X-d
.•.5?-X
cAmEstres
cAmestrEs
cElArent
cAmmEs
|l5
J-K
l>-j
X-d
.-.N-X
cAmEnes
cAmestrE8
cAmEstres
cAmenEa
16
^^-j
&-J
X-d
/.x-x
cEsAre
cAmestrEs
cAmEnes
cAmenEs
nix-d &-J
1
.-.x-x
cElArent
i
cAmestrEs
50
LOGIC AS A PUKE SCIENCE.
Synopsis of all Possible Valid Forms
Sbkies op Syllogisms
in which the middle
Premise is Minor,
AND THE MaXIMUS
Pr/^mise Major op
tus first.
cMArent
e. A, O.
cElArent ferlO
cElArent ifelAptOn I
or, E, A, o ookArdO j"
cElArent ferlsO
fErlo bokArdO
fElApton
fErIm
bokArdO
bokArdO
hOkArdo bokArdO
fEsApo bokArdO
p
§
Regressive Configukation.
frEsIson
cEsAre
cEsAre
cEsAre
or, E, A.o
cBsAre
fEstIno
bokArdO
i. A, 0.
1st fig.
ferlO
fdAptOn I
bokArdO )
ferlsO
bokArdO
18
19
20
211
I
22:
23
24
25
m
27
28:
29
30
31
32
Maximus
Premise.
J-X
J-X
^-X
J-X
&-X
B^-X
d-^X
X-D
X-D
X-J
X-J
X-J
X-J
X-J
Middle Magnus
Premise. Premise.
D-j j X-(l
I
D-
D-
D-
d-
D-
d-
D-
d-
D-
D-
D-
D-
d-
11 — d
D-11
d-n
D-n
J-n
J-n
J-n
J-n
J-n
X-d
n-d
Ultima.
Series op Syllogisms
IN WHICH THE MlDDLE
Premise is Major,
AND the Maxim ls
Premise Minor of
the first.
.-. n-^X
(2)(i) j (3)(4)
I
.*. n— v^X cAniEnes' festlnO
.'. n—^X. c^»iBnes fesApO
\ jl—^^\\cAmEn€8 fresIsOn
:. n-^X
.-. n-^X
/. n-^X
.-. n-^X
.-. n-^X
/. n-s^X
/. n^X
cAmEstres
e,A, O.
fis.
n— v^X cAmEstres festInO
D — 11 I.*. l\—^\'\cA7nEstres fesAvO
d — n I .*. n-wX I \cAmEstres fresis fm
D-n /. n-^Xii
SORITES.
61
of Simple Categorical Sorites. (Continued.)
Series of Syllogisms
EN' WHICH THE MrDDLE
Precise is Mdjor,
i
5
u,
o
1
18
Progressive Contiguration.
'! Series of Syllogisms
j IN which the Middle
Premise is Major,
AND THE MagNITS
Premise Major or
THE FIRST.
Magnus
Premise.
Middle
Premise.
Maximus
Premise.
Ultima.
^ AXD the MagIOTS
j Premise Minor of
I THE first.
1
brAf/iAntip
fErisO
N^-d
D-j
J-X
.-. n-^X
1 (2) (1)
J bArbAra
ioT,A,A,t
(3) (4)
E,a, 0
fEriO
(ITrnAr'is
fEHsO
19
n-d
n-j
J-X
.-. n-^X
dArR
fEriO
(lArApfi
fErisO
20
D-n
D-j
J-X
.-. n-^X
dAvApti
fEriO
d Is Amis fErisO
21
d-n
ii-j
J-X
.-. n-^X
dAtM
fEHO
(lAtlsi fErUO
22
D-n
'i-j
J-X
.-. n-^X
dig Amis
fEnO
bArbAra /ElaptOn 1
or, A, A, i fErisO 1
23
J_n D — j
D-X
.'. n-wXi brAmAntip
fEHO
dArIi fErisO
24
J-n
d-j
D-X
.: n-^X
dim Alia
fEHO
bArbAra bOkardO
25
J-n
D-j
d^X
.-. n^X
1
bArbAra fE^apO \
or, A, A, i frEsisOn f
26
J-n
l>-j
X-D
.*. n-^X
brAmAntip
fEstinO
dArR frEsisOn,
27
J-n
d-j
3t-D
.-. n-^X
dImAris
fEstinO
brAmAntip frEsisOn
28
N— d
D-j
X-J
/. n-^X
1
1
J bArbAra
\oT.A.A.i
E,a, 0
fEstinO
dim Arts
frEsisOn
29
n-d
D-j
X-J i
.-. n-^X
dArR
fEstinO
'! U'Apfi irEsisOn \
\
30
D-n
D-j
X-J
.-. n-^X
dArApti
fEstinO
I
'UsAmis frEHsOn
31
d-n
D-j
X-J
/. n-^X
dAtIsi \
fEstinO
1
dAilH frEsisOn
1 i
32
D-n
d-j
5-.JI
.-. n-^X
dIsAmis
1
fEstinO
52
LOGIC AS A PURE SCIENCE.
Synopsis of all Possible Valid Forms
Series of
IN WHICH 1
Premise
Syllogisms
rHE Middle
IS Minor,
Maximus
Major of
Ph !
O 1
m
33
Kbgressive Configuration.
Series of
IN WHICH
Premise
AND THI
Premise
THE FIRS
Syllogisms
THE Middle
IS Major,
AND THE
Premise
THE FIRST
Maximus
Premise.
Middle
Premise.
Magnus
Premise.
Ultima.
: Maximus
Minor of
r.
^AmEnes
j e. A, 0. \
1 1st fig. f
X-d
&-J
^-}
.-. n-^X
' (2)(1)
cElArent
(3) (4)
i e. A, 0.
1 adfio;
cAmEnes
ferlO
34
X-d
B--J
n-j
.-. n-^X
cElArent
festInO
cAmEnes
or, A, E,o
felAptOn )
35
X-d
^-J
J-n
/. n-^X
cElArent
fesApO
xAmEnes
ferlsO
36;
X-d
B-J
j-^i
.-. n-^X
CElArent
fresIsOn
37
X-d
l)-j
5^-J
/. n^X
bArbAra
j a, E, 0.
1 2d fig.
38
X-d
D-J
n-wJ
/. n-^X
bArbAra
bar Ok 0
39
X-d
D-J
J-X
/. n^X
bArbAra
( a, E, 0.
» 4th fig.
cAmEstres
1 let fig. f
40
X-j
^-J
X-d
/. n-^X
cEsAre
J e, A. 0.
1 2d fig.
cAmEstres
ferlO
41
X-j
&-J
n~d
.-. n-^X
cEsAre
feotInO
cAmEstres
or, A^E^o
felAptOn 1
bokArdO f
42'
X-j
^-J
D-n
.-. n-^X
cEsAre
fesApO
cAmEstres
ferlsO
43
x-j
i>-J
d-n
cEsAre
free Is On
hArOko
bokAr.iO
44
x-j
I
d-^J
D-n
.-. n-^X
1
SORITES.
53
of Simple Categorical Sorites. (Concluded.)
Series of Stixogisms
IN WHICH THE MIDDLE
i:
Progressive Configuration.
Series op Syllogisms
IN WHICH THE MIDDLE
Premise is Mijjob,
o
Premise
AND TH
IS Major,
AND THE Magnus
E Magnus
Premise Major of
Tj *■
Magnus Middle
lilaximus
Ultima.
Premise
Minor op
THE FIRST.
O
^ ■
33
Premise. Premise.
Premise.
THE first.
cAmEstres
j A\ e, 0. ]
1 4th fig. f^
i
N-j B--J
X-d
j
J cEsAre
1 or, E, A. 0
^3) (4)
A,e, 0
bArokO
1
U
n_j 1 &- J
X-d
.-. n-^x;
fEstIno
bArokO
1
1
35
J_n &-J
x-d
.-. 11-^x'
fEsApo
bArokO
30
j_n B--J
X-d
.-. n_Xi
frEsIson
bArokO
cEsAre
S A, e, 0.
\ 4th fig.
37
5^_J D-j
x-d
.-. ii-^X
J cAmEstres A. e. 0
j or, A.E,o bAjvkO
38
n-^JjD-j
x-d
.', n^X
hArOko
bArokO
cMArent
\ A, e, 0.
\ 4th fig.!
39
J-N
D-J
x-d
.-. n-s^X
j cAmE'ies
\ or, A,E,o
A, e. 0
bArokO
cAmEnes
) A, e, 0.
\ 4th fig.
40
N-d
&-J
x-j
.-. n-^X
J cElArent
\ or, E,A.o
A.e. 0
bArokO
41
n-(l
B--J
x-j
.-. n-wX
fErlo
bAwkO
1
42
D-n
&-J
x-j
.-. n-^X
fElApton
bArokO
43
d-ii
]&-J
x-j
.-. n-^X
fErleo
bArokO
j
44
D-n
d-^J
x-j
••— "
bOkArdo
bArokO
64 LOGIC AS A PURE SCIENCE.
§ 11. The number of forms of valid Sorites, shown in
the foregoing synopsis, is eighty-eight, forty-four on each
side ; but a comparison of them, line by line, read across
both iDages of the synopsis, will show that, considered
with respect to the propositions of which they are com-
posed, without regard to the order of their statement,
there are but forty-four ; the first and third propositions
in the regressive configuration changing places in each
case, and becoming respectively third and first in the
progressive throughout the whole series, the middle
premise and ultima being the same in each case on both
sides throughout. They are numbered from one to forty-
four, on each side, to correspond.
To one or another of these forms, EYElfY valid argu-
ment (expressed categorically) involving four terms, or,
as will be hereinafter shown, involving any greater num-
ber of terms, MUST BE conformed.
§ 12. The moods, as determined by the quantity and
quality of the propositions (indicated by their symbols),
are twenty in number, of which fourteen occur in both
configurations, three in the regressive only, and three in
the progressive only.
The following table shows them, arranged in the order
A, I, E, O of the symbols of the ultima, with their num-
bers in each configuration, as in the synopsis, repeated
where they are both minor and major. The symbols are
in capitals in the synopsis, the first two in the columns
of Syllogisms, on the right-hand side of each page
{majors) being transposed, as previously stated, and as
shown by the figures over those columns.
SO«ITES
55
Moods of Sorites.
1 i
Nos. m Regressive Conpigubation, Nos. ts Progbessite Configuration.
Symbol?.
ij
Minors.
Majors. J Minors. Majors.
A, A, A, A.
1.
1.
A, A, A, I.
2, 4, 7, 10.
4, 7, 10.
2, 4, 7, 10.
2, 4, 7.
A, A, I, I.
3, 5.
9,11.
1, A, A, I.
9. 11.
9,11.
3,5.
3,5.
A, I, A, I.
6,8.
6,8.
6, 8.
6,8.
E,A,A,E.
12, 13.
12, 13.
14, 15.
14, 15.
A, A, E, E.
14, 15.
12, 13. ^
16, 17.
A, E, A, E.
16, 17.
16, 17.
16, 17.
E, A, A, 0.
J 18, 20, 23, 1
|26, 28, 30. j"
18, 20, 28, 30.
37, 39.
37, 39.
A, A, E, 0.
37, 39.
jl8, 20, 23, (
126, 28, 30. i
j 18, 20, 23, )
126, 28, 30. (■
E, A, I, 0.
19, 21, 29, 31.
19, 21, 29, 31.
I, A, E, 0.
19, 21, 29, 31.
19, 21, 29, 31.
E, L A, 0.
22, 24, 27, 32.
A, I, E, 0.
22, 24, 27, 32.
22, 24, 27, 32.
0, A, A, 0.
25. •
38.
A, A, 0, 0.
38.
25.
A, E, A, 0.
33, 35, 40, 42.
33, 35, 40, 42. 1
33, 40.
33, 35, 40, 42.
A, E, I, 0.
34, 36, 41, 43.
34, 36, 41, 43.
I. E, A, 0.
34, 36, 41, 43.
A. 0, A, 0.
44.
44.
66 LOGIC AS A PURE SCIENCE.
It is manifest that it would be a very difficult thing
to classify Sorites in figures, according to the positions
of the terms, and to devise names for the moods, anal-
ogous to those of simple Syllogisms ; and, if it should
be accomplished, the figures and names of the moods
would be extremely burdensome to the memory. The
different forms can be much more readily referred to
by their numbers and the names of the configurations,
as adopted, than by their symbols, or any names that
could be devised for them. They will be hereinafter so
referred to.
By counting the series of Syllogisms on the left
{minors) and right {majors) of the synopsis in each con-
figuration, there will be found to be :
J^ Regressives. Progressives.
Minors. Majors, Minors. Majors.
39. '62. 33. 40.
corresponding to the numbers shown in the table on
page 46.
§ 13. Sorites, in the regressive configuration, may be
expanded into series of Syllogisms in all combinations of
figures, except those of the third and first, and third and
second ; and those in the progressive configuration, in all
combinations, except those of the second and first, and
second and third.
Such of them as can be expanded wholly in the first
figure, are the only perfect forms. The series of Syllogisms,
in which they can be so expanded, occur in the synopsis
only on the left side of the regressives {minors), and on
the right side of the progressives {majors) ; and the first
SOEITESo 57
figure occurs as the figure of the second Syllogism only
on the same sides. Moods Nos. 10, 11, 15, 26, 27, 35,
36, and 39 cannot be expanded directly (that is, without
conversion) except by the aid of the fourth figure ; a
fact which may tend in some measure to relieve that
figure from the odium which has been cast upon it.
§ 14. Tliere is a very remarkable and wonderful anal-
og}^ between the forms of reasoning and the two simplest
forms of geometrical figures, plane and solid (with plane
surfaces) ; an analogy which is evidently something more
than merel}' fanciful.
The Syllogism of logic and the triangle of geometry,
and the Sorites and tetrahedron are, respectively, similar.
The triangle consists of three points, equivalent to
the three terms of the Syllogism, connected by three
lines, which answer to the copulas of the propositions.
l\o plane surface can be represented by less points and
lines, no argument by less terms and propositions. By
means of the former, with the aid of the latter, all phys-
ical relations in space are determined, not only on the sur-
face of the earth and within it, irom those of the smallest
subdivision to those of continents and oceans, but also in
the heavens to the remotest star-depths, so far as the
stars can be brought under observation ; by the latter
all relations are determined, not only of physical things,
but also of the metaphysical and immaterial. But the
analogy does not end here. In its very practical con-
struction the triangle produces the equivalent of a per-
fect Syllogism in Barbara. If we are at any point, N,
on the surface of the earth, from which we can see
68 LOGIC AS A PURE SCIEISCE.
another point, J (either on the earth or in the heavens),
which is inaccessible, and the distance to which we can-
not therefore directly measure, we may select another
point, D (either on the earth or its orbit), which is acces-
sible, and from which the point, J, may also be seen ;
and first, carefully observing the directions from N to J,
and from N to D, and thus determining the angle, we may
then proceed to measure the distance between N and D
in a straight line. The line thus laid down is equivalent
to the first proposition, IS" — d, with which we set out in
§ 2 of this chapter. Arrived at D, we may then observe
the direction from D to J, and determine the angle, and
then, by means of the elements thus obtained, we may
determine the distance in a straight line from D to J, and
from N to J. The lines thus drawn, or supj)osed to be
drawn, are the equivalents of the second proposition,
D — j, with which we set out, and of the conclusion to be
deduced from it and the first proposition, IS" — d, when
put forth as premises of a Syllogism, namely, 'N — j.
The tetrahedron is the simplest form in which any
solid with plane surfaces can be included, and is the
analogue of the Sorites. Its four points answer to the
four terms, its four planes (each in the form of a triangle)
bounded by six lines (each being a boundary of two
planes) to the four Syllogisms of the two principal
series ; each series with its six propositions. Each plane
connects three points, each Sjilogism three terms. Each
of the four points is excluded from one of the planes,
each of the four terms from one of the Syllogisms.
To illustrate by means of geometrical figures :
If we take a piece of card-board and, having cut it in
SORITES. 69
the form of an equilateral triangle, inscribe therein
another equilateral triangle, the lines of which terminate
in the middle of the lines of the exterior one, and mark
all the angles with letters, as follows :
we may then fold the card-board backward on the lines
of the inscribed triangle so as to bring together the three
points, X, X, X, and then fastening together the edges
of the card -board so brought together, we shall have a
regular tetrahedron, the very embodiment of a simple
Sorites. Looked at from our present stand-point, we
shall see only the inscribed triangle No. 3, and having its
angles marked with the letters N, D, and J. The other
triangles and the point X will not be seen. Turning the
figure about, so as to bring its planes before us in the
order in which they are numbered, and considering them
in two series of two each, we shall find them as follows,
beginning at the right hand with the first series, and
reading backward, but from left to right, in the second.
60
LOGIC AS A PURE SCIENCE.
First series.
Second series.
Observing that the letters at the apices of the tri-
angles are the middle terms of the Syllogisms of the two
principal series hereinbefore shown, and considering the
lines of the triangles as copulas connecting the terms of
propositions, and the lines at the bases as indicating con-
clusions, and beginning with the first series of triangles
at the right hand and regressing^ we can read as follows :
Because D is J and J is X, therefore D is X ; and because N is D and
D is X, therefore N is X,
and then going to the second series, and beginning at the
left hand and progressing, we can further read :
Because N is D and D is J, therefore N is J ; and because N is J and
J is X, therefore N is X.
The correspondence between the triangles and the
Syllogisms is exact throughout, except that the premises
SORITES.
61
in the latter are transposed, but the order of statement
of the premises is a matter of no consequence, the terms
determining their character.
The middle terms D and J may, of course, be tmns-
posed in our original illustration, and in such case the
numbers 2 and 4 would also have to be transposed, and
the i:)ositions of all the letters and the numbers in tri-
angles 1 and 3, relatively to the whole figure, would
also require to be changed. The first series of triangles
would then read forward and the second backward, but
the series of Syllogisms would remain the same, the
first regressive, and the second progressive.
The four triangles may also be exhibited in the
following form :
and may be folded on the interior lines with like result
as before.
But the Sorites is superior to its analogue, the tetra-
hedron, in this, that its ultimate conclusion is reached by
either process, regressive or progressive, but both are
required to complete the tetrahedron: This will be
apparent by the consideration of the two following
forms.
62
LOGIC AS A PUKE SCIENCE.
/ \
/ N
If, in the first, beginning with 'N, we successively
reach by investigation the points D, J, and X, and then
commence to reason with the propositions which we
enounce as the results of our investigation, we may by
two Syllogisms, of which the two completed triangles
1 and 2 are analogues, arrive at the ultimate conclusion.
But if, in the second, by the same process of investiga-
tion we reach only to the point J, and then commence
SORITES. 63
to reason, we frame our first Syllogism, of which the
triangle 3 is the analogue, resulting in the conclusion
that N — J. We are thereupon, if we would advance^
further, obliged to resume investigation, and through it
reach out to X, and are thence enabled to frame the
second Syllogism, of which the triangle 4 is the analogue, •■
arriving at the same ultimate conclusion. But in either
case the tetrahedron is incomplete, and can only be com-
pleted by the union of the two. Each figure is the com-
plement of the other, required to make the perfect figure,
shown in our first illustration.
But again, the two different processes, regressive and
progressive, in respect to argumentation by Syllogisms,
are analogous to the two possible combinations of the
two processes by which we may determine the length of
the concluding line with which we enclose a triangle.
Leaving N, and going to D, we observe the direction
in which we are traveling, and measure the distance
tmveled. Then observing the direction from D to J,
and thus determining the angle, we go on from D to J,
measuring the distance. If we then stop, we may, by
the three elements thus obtained, viz., the two lines and
the included angle, determine the distance and direction
from N to J ; then, having this distance and direction,
and observing the direction of X from J, we go back to
N, and observe its direction from X, and determine the
angles, and then with the three elements thus secondly
obtained, viz., the two angles and the included line from
N to J, we may determine the distance from N to X.
This is analogous to the progressive process.
But if, after reaching J, without stopping to de-
64 LOGIC AS A PURE SCIENCE.
termine its distance from 'N, we observe the direction
therefrom to X, as in triangle 1, and going back to D,
observe also its direction from X, and determine both
angles, then with the three elements thus obtained (being
like to those of the second three in the preceding
process), we may determine the distance from B to X,
and then, having the distances and directions from D to
IS", and from D to X (the figure being now considered
as folded), and determining the included angle, we may
by such elements (being like to those of the first three
in the preceding process) determine the distance from
N to X. This is analogous to the regressive process.
Surely, in all this wonderful accord there must be
something more than mere coincidence. ''The invisible
things of God are clearly seen, being perceived through
the things that are made."
§ 15. But the subject concerning which we set out to
make investigation may be the summum genus instead
of the infima species or individual, as hitherto, and in
such case we shall find that the processes of both inves-
tigation and reasoning will be in the exactly opposite
direction, and that the maximus term, instead of the
magnus, as hitherto, will be the subject of the ultima,
and the magnus term instead of the maximus will become
the predicate.
Strictly speaking, the word "predicate" is not prop-
erly applicable to the last, but rather to the first term of
propositions as they will be exhibited in this section,
inasmuch as the species cannot be predicated of the
genus, but the genus of the species. To change the
SOEITES.
Qd
names of the terms as they stand related to the proposi-
tions would, however, be confusing, and they \vill, there-
fore, be retained in their grammatical rather than in
their strict, logical signification.
But we shall find it necessary to change the signifi-
cation of the copula. As hitherto employed, such sig-
nification has been '*is" or '"is not" in the sense of ""is
(or is not) coinprehended in,^^ but as employed in this
section ohly, the copula must be understood to signify
''comprehends^'^ or ''does not comprehend.'^ The rea-
son for this change, if not immediately obvious, will
become clear as we jjrogress. It will, however, be here-
after seen that in some cases the two significations are
interchangeable, and either may be understood.
I shall have immediate recourse to illustration by
means of geometrical figures, as thereby such illustmtion
can be made much clearer, being exhibited to the eye as
well as to the understanding ; and I now give the fol-
lowing fiarure.
66 LOGIC AS A PURE SCIENCE.
which you will observe is like to our original card-
board figure on page 59, with triangles 1 and 3 remain-
ing in the same position as therein, but with triangles
2 and 4. turned upward, each in a semicircle, on the
points D and J as centres respectively.
The jioints XXX are now brought together in the
figure, and N JS" N separated and become exterior. The
X)oints D D D and J J J retain their intermediate posi-
tions.
If now we begin to make investigation concerning X
as the subject, we shall find ourselves proceeding in a
descending instead of ascending direction, as before ; and
we shall also find that the notions which we discover as
predicable (in the sense of the copula, as above changed),
of our successive subjects, instead of being higher genera
and comprehending the subjects, are lower species, and
are wholly comprehended in the subjects respectively.
The propositions in which we lay down our judgments
will therefore necessarily be required to signify this dif-
ference, which may be done by j)utting the predicates in
capitals instead of small letters, as before, and will be as
follows :
X ■: — J {meaning All X comprehends all J) ;
J — D {meaning All J comprehends all D) ;
D — N {meaning All D comprehends all N).
The subject of each of the foregoing propositions is
distributed. But it might have been undistributed in
so far as relates to the manner of its representation, and
the proposition still retain its character as universal.
To illustrate, I now reproduce the first combination of
circles shown in the former part of this treatise when
SORITES.
67
treating of simple Syllogisms, adding another circle to
make it applicable to a Sorites, the letters being put on
the lines of the circles, and to be considered as indicating
the whole areas included in the circles respectively.
It will now be manifest, from mere inspection of the
figure, that what we have predicated of X (viz., J) might
also have been predicated of x, and in fact with more
correctness, for J is comprehended wholly and only in
tliat part of X which lies within the circle marked on
one side J and on the other x. In like manner, what we
have predicated of J (viz., D) might have been predi-
cated of j, and what we have predicated of D (viz., N)
might have been predicated of d.
The propositions may therefore be stated as follows :
X or X — J ;
J or j — D ;
D or d — K.
In either alternative the propositions must be re-
garded as universal. I shall hereafter make use only
68 LOGIC AS A PURE SCIENCE.
of that in which the subjects are represented by small
letters, as apparently, but not in fact, undistributed. In
reading the propositions, the words "All" and ''Some"
must be expressed, and it must be borne in mind that'
the word "Some" applies to a definite part of the
term, and when in the process of the reasoning a tenii
with that word prefixed shall be repeated, it must be
read or understood as "The same some," or "The
same definite part of."
The dictum of Aristotle, as applicable to the above
propositions, will now have to be changed so as to read
as follows.
Whatever definite term is afiirmed or denied as com-
prehending any other definite term, may be afiinned or
denied as comprehending any definite term compre-
hended in the definite term so comprehended, and in
like manner of any definite term comprehended in the
definite term so secondly comprehended, and so on ad
infinitum.
Applying the dictum as thus changed to the above
propositions, the two forms of the full Sorites warranted
thereby will be as follows :
In the regressive
configuration.
In
the progressive
configuration.
d — N,
X- J,
j - D,
J-D,
X- J;
d — N;
.-. X — N.
•
•. X - N,
and the abridged form will be
••• j —
D; .-.
X -
- K.
SOKITES. 69
All propositions put forth in the above form in the
descending processes of investigation and reasoning,
may be converted simply, provided the original signifi-
cation of the copula be at the same time reinstated, and
by simple conversion of the above, we shall have the two
forms of Sorites as we have hereinbefore seen them.
But not only have the terms of all the propositions in
the two forms changed places, but also the forms them-
selves, in respect to the configurations, the converse of
that which before was regressive having become progress-
ive and of that which was progressive, regressive. By
examining our original card-board figure in connection
with the figures on page 62, and the remarks on the lat-
ter, and comparing them with the fii-st figure in this sec-
tion, and applying such remarks to the configurations as
herein gi'ven, it will be seen that such change is proper,
triangles 3 and 4 in the latter figure being the analogue
of the Sorites in the regressive configuration, and 1 and
2 of that in the progressive.
In like manner, it will be found that in all matters of
form there will be continued inversions.
The Sorites herein given may be expanded into series
of Syllogisms as follows :
In the regressive process.
j - D, X - J,
d-N; j-N;
.-. j — N. --^^^^ .'. X — N.
In the progressive process.
X — J, ^ X — D,
j - D ; X^ d - N ;
.-. X — P -^ .'. X — N.
70
LOGIC AS A PURE SCIENCE.
All the propositions in the foregoing forms are uni-
versal, but they may all be particular in the manner of
their representation (indicated by the apparent non-distri-
bution of the predicate), provided the definiteness of the
terms represented be kept in view. Thus, in the fol-
lowing figure, let the letters on the lines of the circles
refer to the whole areas of the circles respectively as
before, and those in areas only to the areas as bounded
by lines respectively, but considering them where occur-
ring more than once as to be taken together :
The Sorites exemplified will be as follows :
In the regressive
In
the 2)?'ogressive
configuration.
configuration.
d — 11,
X — J,
J - d,
J - d,
X — j;
d - n ;
.'. X 11.
.-. X — 11 ;
and may be expanded into series of Syllogisms, as fol
lows:
SORITES. 71
In the regressive process.
j — <^, X — j,
a — n ; ^ j — n ;
.-. j — 11. .*. X — n.
In the progressive process.
X — j, ^ X — d,
j — d; ^y^ d — n;
.-. X — d. — ^'^'"^ .*. X — 11.
Here apparently we have two anomalies — viz., Syllo-
gisms having the middle terms undistributed in both
premises, and Syllogisms in which conclusions are de-
duced from particular premises. But they are such
only in appeaiTince ; all the propositions (the definiteness
of the terms being kept in mind) being in fact universal,
and the middle term distributed in each case in the
major premise.
The teiTiis of all the foregoing propositions may
each be considered as comprising all the areas marked
in the figure ^\'ith the small letters representing them
respectively, taken together respectively^ or only those
areas respectively, in which the letters representing
both the subject and predicate appear, taken together.
In the former case, the subjects will each be greater
than their predicates respectively, and the copula must
signify ''^ comprehends,'" in the latter, the terms of
each proposition will be co-extensive, and the copula
may have either signification. But in the latter case, the
major middle term in the middle premise will narrow in
signification to that of the minor-middle term, and the
maximus term in the ultima will have a narrower sig-
nification than as employed in the maximus premise.
72 LOGIC AS A PURE SCIENCE.
The middle premise, it will be seen, has become the
major premise, and. the magnus premise the minor of the
first Syllogism of the series in the regressive process,
and the middle premise has become the minor, and the
maxim us the major of the first Syllogism of the series in
the progressive. The regressive Sorites in the descending-
process is therefore, in the forms above given, which you
will find are the perfect forms, a major Sorites, instead of
a minor, and the progressive a minor Sorites instead of a
major as before. It will be also seen that the Enthymeme
taken from the second Syllogism in the regressive series
is of the second instead of the first order as before, and
vice versa in the progressive. If a synopsis should be
made, this would necessitate (to make it conform to the
former) the transfer of the headings of the columns on
each side of each page of the former from each page to
the other, and their transposition after being so trans-
ferred.
All the Syllogisms are in the fourth figure, which in
this process becomes the perfect figure, the first be-
coming imperfect. The second and third figures will
also be found to have changed places, if indeed they
and the first can have any place at all, in the new
sense of the copula. One of the premises in each case
in the second and third figures, and both in the first,
would necessarily be in the inverse order, affirming or
denying of the species that it comprehends or does not
comprehend the genus, or else the original signification
of the copula would have to be considered as reinstated
in such premises, and the process would thereby lose
its distinctive character as a process wholly in the
SOEiTES.
descending direction, which only we are now considering.
By examining the synopsis, it will be fonnd that in all
cases in which either of the involved Syllogisms in the
columns on the left side of the regressives or right side
of the progressives is in one of the imperfect figures,
and in all cases of combinations of Syllogisms shown
on the other side of each page respectively, the process
of the reasoning partakes of both characters, being partly
in the ascending and partly in the descending direction.
I shall not proceed further with the consideration
of this subject, for the reason that propositions in the
descending process are seldom, if ever, put forth in form
as herein given, but in the converse. When you come to
the study of Logic as illustrated by concrete examples
(in which aspect it is, in respect to each such illustration,
an applied science), you will find a distinction made in
respect to the quantity of concepts (terms) as being either
in extension or intension, the latter being called also
comprehevsion. This distinction runs also into the prop-
ositions and syllogisms as treated of, according as the
terms are considered as in one or the other quantity.
You will find it, however, to be of no practical impor-
tance in so far as the process of reasoning is concerned,,
all reasoning being conducted on the lines of the pro-
cess, as we have previously considered it, and being
called reasoning in extension, in contradistinction to the
process as shown in this section, which is called reason-
ing in intension or comprehension. The distinction, in
so far as it relates to the terms (concepts), does not lie
within the province of Logic as a Pure Science, and
cannot be illustrated by means of symbols indefinite
74 LOGIC AS A PUEE SCIENCE.
in material signification, but the illustration of the pro-
cesses of investigation and reasoning wholly in the
descending direction, given in this section, will serve to
make it, as continued into the reasoning process, clearer
and more easily understood.
The consideration of the subject matter of this sec-
tion would perhaps have been more appropriately intro-
duced when treating of simple Syllogisms, but it could
not have been made as intelligible without as with geo-
metrical illustration by combinations of triangles, and
the latter has been more apj^ropriately, and at the
same time more effectively, introduced in this chapter,
where it has been exhibited in one view, and to its full
extent.
The copula must now be considered as returned to
its original signification, and where the word " descend-
ing" shall be hereinafter used, it must be considered
as applicable to the direction of the process of inves-
tigation, but not to the form of the propositions, which,
in the perfect moods of the Sorites, will always be found
in the converse of those herein given.
§ 16. Thus far the premises of the Sorites exhibited
have consisted of propositions put forth independently
as the results of investigation. They may, however, be
the results of prior processes of reasoning, the premises
of which may be required to be exhibited in connection
with them, in order to a clear understanding of the prin-
cipal argument. The full expression in such case will
become complex, and may be in two forms, of which I
first exhibit the following:
SORITES.
J - X,
*.' Z — X and Y — z and J — y.
D-J.
•.• B - j and D - b.
N — a.
•.• K-d and N-k;
N— X.
•
7a
Here each premise is the ultima or conclusion of a
i^rior process of reasoning, the premises of which are
affixed^ with the word "because'' preceding.
In the example, all the premises have supporting
X^remises affixed. But any one, or two, only, may have
such premises affixed, the other two, or one, as the case
may be, being propositions put forth independently.
The whole expression, in either case, is called an
Ex^icheirema, or Reason-rendering Syllogism (of either
three or four temis). The principal argument, with ref-
erence to the supi)orting premises, is called an Episyllo-
gism ; and the supporting premises in each case, ^vith
reference to the premise proved, is called a Prosyllogism.
The second form is that in which the premises of the
Prosyllogism are prefixed^ those in relation to the first
premise being stated antecedently to the whole principal
expression : those in relation to the second or middle
premise, interpolated between the first and middle, and
those in relation to the last, interpolated between the
middle and last.
If either of the first two be in such form, it will be
found ui)on trial, that the principal expression has lost
in forcibleness of statement or in perspicuity, and they
may, therefore, be disregarded, but the third \\ill be
found to lead to greater perspicuity, and especially so if
more than two new middle terms are called into requi-
sition for the purpose of elucidation.
76 LOGIC AS A PUKE SCIENCE.
The first form (Epicheirema) is better adapted to the
statement of arguments in which the premises are ex-
plained, the second to those in which either the first or
last premise is disputed. It is seldom the case in any
disputation, that more than one of the premises of the
principal argument is called in question, and that one
is generally the first or last, the middle premise being
usually a general rule acquiesced in upon being stated ;
and if the disputed yjremise be the first, the principal
argument, by changing the configuration, may be thrown
into such form that it shall become the last.
I now proceed to consider Sorites as complex exj)res-
sions, in the second form, but only as limited to those in
which the last premise is disputed, and to distinguish
them as such, shall call them Compound Sorites.
§ 17. A Compound Sorites, once compounded, tcTien
fully expressed, consists of a simple Sorites (herein
called the principal Sorites) with two, or three, proposi-
tions interpolated between its middle and last premises ;
such propositions (if there be two) constituting the pre-
mises of a simple Syllogism of which the conclusion, or
(if there be three) of a simple Sorites, of which the
ultima is the last premise of \\\% principal Sorites. The
interpolated propositions will be herein called the in-
cluded Enthymeme, if there be two, or Sorites, if
there be three, giving the full name in the latter
case, in default of one analogous to Enthymeme in the
former. An included Sorites may in like manner
have an Enthymeme or second Sorites included within
it, and the second included Sorites may in like manner
have an Enthymeme or third Sorites included within it,.
Ed
SORITES. 77
cind so on ad infinitum. There can be but one included
Enthymeme, and it will always be the last included ex-
pression. The reasoning in all such cases, while it will
ve the appearance of being very much involved, will
in reality be very much clearer.
§ 18. But compound Sorites are seldom, if ever, f^llly
expressed informal, prepared argumentation, the last
premise of the principal Sorites being suppressed, but,
as will be hereinafter shown, in all cases implied. In
this aspect a compound Sorites may be better defined
as an argument consisting of more than four expressed
propositions composed of as many terms as there are
expressed propositions, including the ultima. Both
definitions Avill be better understood by illustration.
Let us suppose the case of two disputants of whom
one, the proponent, advances these propositions :
I> - J:
.-. X - X,
to which the other, the opponent, answers : I admit that
D — j, but I deny that it follows that X — x.
The propositions, as you will observe, constitute the
abridged form of the first mood.
The proponent replies, asserting, as the reason, the
two i)ropositions necessary to make up the expanded
form, viz.:
•.• J — X
and X — d,
and to this the opponent makes rejoinder: Admitting
that J — - X, I deny that X — d.
78 LOGIC AS A PURE SCIENCE.
The issue is now clearly defined, and the whole case
may be stated as follows :
J — X
admitted,
I)- j
admitted.
X — d
alleged but denied.
X — X
claimed but denied.
The proponent, to maintain the issue on his part,
must establish that 'N — d, or must fail.
To do it, as the proposition to be established is A, he
must find a middle term, with which both the terms 'N
and D may be compared, so as to form, with the con-
clusion, a perfect Syllogism in Barbara (symbols AAA),
or two middle terms, with one of which IS" may be com-
pared and D with the other, and one of which may be
predicated of the other, all in such manner as to con-
stitute, with the ultima, a valid Sorites in the first mood
(symbols A AAA).
Let the middle term, in the first case, be Y, and the
two middle terms, in the second case, be Y and Z.
The Syllogism in the first case will be :
Y- d,
N - y;
.-. N — d.
But in the second case the two new terms are required
to be compared, and either may be the subject of the
proposition in which they are compared, viz., Y — z or
Z — y. The abridged Sorites may therefore be either :
Y — z ; .-. N — d ;
or, Z - y; .-, N - d.
I
SORITES. 79'
Let us take the first, and in order to expand it into
a full Sorites, let ns write down the first mood in the
regressive configuration, as in the synopsis, and write
under its second and fourth propositions the abridged
Sorites thus taken, as follows :
J — x; D — j; N — d; .-. X — x.
Y — z; ,-. :N^ — d.
Then^ by expressing the first and third implied
propositions of the abridged Sorites (making them to
correspond in respect to the terms employed), we shall
have the expanded Sorites as follows :
Z — d ; Y — z ; N — y; .-. N — d.
By taking from the Syllogism in the first case its two
premises (constituting an Enthymeme of the third order),
and from the Sorites in the second case, its three pre-
mises, and interpolating them (respectively) between the
middle and last premises of the principal Sorites, w^e
shall have, in each case, a compound Sorites fully ex-
pressed, as follows :
In the first case.
In the second case.
J - X,
J — X,
D- j,
D- j>
Y-d,
Z -d.
N-y;
Y- 2,
.-.
N — y;
N — d.
.•.
and .•• N — x.
N-d,
and .-. N — X.
60 LOGIC AS A PURE SCIENCE.
The conclusion of the first Enthymeme of the princi-
pal Sorites, viz., J) — x, is held in the mind ready to
unite with the last premise, N — d (after the latter shall
have been proved), in establishing the ultima, N — x.
§ 19. But there is a shorter and simpler process, and
the one which is usually employed in formal, prepared
argumentation. Instead of holding in the mind the con-
clusion of the first Enthymeme to unite with the last
premise of the principal Sorites when proved, as above
stated, we may at once employ it (mentally) as a premise
in connection with the first of the new expressed proposi-
tions, and in like manner the unexpressed conclusion re-
sulting from them as a premise in connection with the
second new expressed proposition (and in the second case
as above, the unexpressed conclusion thus resulting in
connection with the third), and shall find that the last
premise of the principal Sorites will not appear. Thus,
in the two cases, the unexpressed conclusions being given
in italics :
In the first case. In the second case.
J — X, J — X,
D — j, (,.-.D-x). J) — ], (.:D-x).
Y — d, (.-. Y-x). Z — d, (.-. z-x).
N — j; Y — z, (.-. Y-x).
.: K - X. N -y;
.-. N — X.
But the last premise of the principal Sorites will have
been implied, as will be manifest from a comparison of
the two forms in the second case put side by side, as
follows :
soe!tes. 81
First form i)i second case. Second form in second case.
J - X,
J — X.
D — j; {.. B—x, held in the mind).
D- j,
_ _ _ _
Z — d,
Z - rt.
Y - z,
Y- z,
N - y ;
N-y;
.-. N — X.
" X-d;
aud .
•. N — X. (••• D-x).
The second form is the simpler, but the first is the
clearer, exhibiting the entire process of the reasoning.
The included Enthymeme in the first case, or Sorites
in the second, serves only to pi^ove the last premise of
the principal Sorites, and forms no part of the argument^
which is wholly contained in the principal Sorites.
§ 20. Both the principal and included Sorites in the
examples are in the regressive configuration, but they
may be in different configurations. If in the foregoing
disputation the opponent in his rejoinder had admitted
the magnus premise, !N^ — d, but denied the maximus,
J — X, the principal Sorites of the proponent would
have been in the progressive configuration, and the in-
cluded one could have still been in the regressive, viz. :
N — d,
D - i;
Z - X,
Y -z,
J -y;
J X,
and
.-. N — X.
82 LOGIC AS A PURE SCIENCE.
The two configurations cannot be directly linked
together in this example, as before shown in the second
form, there being a break in the chain between the
second and third propositions. But by considering the
configuration of the included Sorites to be changed (as
it may be by transposing the first and third premises
thereof), the whole expression can be put in the second
form as before, and the last premise of the principal
Sorites, J — x, will not appear. It does not, however,
follow that the two configurations cannot in any case be
directly linked together. That they may be in some
cases mil be hereinafter seen.
§ 21. All the Syllogisms involved in all the foregoing
examples are in Barbara, and the dictum of Aristotle,
as. hereinbefore extended, may be directly applied to
those in the second form, by extending it still further in
like manner. But to those in the first form it would
have to be twice applied, first to the included Sorites
and secondly to the principal, and in that case would
not require to be further extended, both the Sorites
being simple.
§ 22. But if any of the involved Syllogisms are in
any other figure, or combination of figures, they would
have to be converted into Syllogisms in the first figure,
before the dictum could be directly applied.
The following are examples of compound Sorites,
the involved Syllogisms of which are in combinations of
figures, as shown by the names of the moods given in
connection with them. The conclusions proved, but not
expressed, are also given in italics in connection with
the names of the moods, except the ultima of the in-
SORITES.
83
eluded Sorites (in the first forms), which is expressed as
a premise below the second dotted line. The principal
Sorites and the number of its mood and the configuration
are given in advance of each example :_
Gth Regressive Mood.
J — X, d — j, D — n ; .-. n — x.
First form.
J — X,
d — j ; (.-. d-x, Dani),
D— Z,
Z-y,
Y-n;
D — n;
and .*. 11 — X. (•.• d—x, DiBomis).
Second form.
J-x,
d — j, (.-. d—x. DarU\
D — Z, (.-. z—x, Di8amis)y
7i — y, (,'. y—x, Disamis\
Y-n;
;. n — X. {Disamis).
15th Pi'ogressive Jlood.
N, D— j, X — d; .-. ^ — X.
First form.
D — j; (A N-D,
Y-d,
z-y,
X — z:
Second form.
^— N,
D — j, (.-. ^—D, Camenes),
Y — d, (.-. N— Y, Came8treg\
Z — y, (.♦. ^— Z, Camestres\
X-z;
*. Jf — X. iCamegtreg).
X-d;
and /. ?^ — X. (:• N-D, Camestres).
84
LOGIC AS A PURE SCIENCE.
25th Regressive Mood.
and
d-^X,
D
- i, J -
— n ; .-. n -^ X.
Nrst form.
Second form.
d— X,
d-^X,
D-j; u.j-
-X,
Bokardo),
D — j, i.'.j-^X, Bokardo\
J — y, {.-.y-^X, Bokardo)
J-y,
Y — Z, (.-. z-^X, Bokardo)
Y-z,
Z— n;
Z-n;
/. n -w X. {Bokardo).
J-n;
n-^X. (vi-
■ X, Sokardo).
% 23. The included Sorites may have an Enthymeme
or a second Sorites included within it, and the second
included Sorites may have an Enthymeme or third Sorites
included within it, and so on ad infinitum. Thus :
First form
Second form.
J-x,
J X,
D — j ; {.-. B — x, held in t?is mind).
D — j, (.-. D-x),
Y-d, (.-. Y-x),
j Y-d,
Z— y, (.-. z-x).
( Z — y; (.-. Z — rf, hdd in the mind).
z — k, (.-. k-x).
1
K — q, (.-. q-x\
i
z— k, ^
K— q,
Q — n;
^ Second included Sorites^
Q-n;
.-. n — X.
( z— n;
( .-. (••• Z-d),
d— n;
ar
id .-. n — X.
C- D-x).
SOEITES.
85
If the first included Sorites in the last example be
put in the regressive configuration, its last premise will
be Y — d instead of z — n, and the second included
Sorites will be employed to establish the former instead
of the latter, but of course by different premises. In
such case we shall find that when we attempt to put the
whole expression in the second form, the premises of the
second included Sorites will take precedence of those in
the first, and the latter wiU be transposed. Thus:
M
First form.
J — X,
13 — J • (.'. D — x, held in the mind),
Z — n,
Z — V ; (.-. y — n, held in the mind)y
K-q
Y — k
Y-d;
i-'y-ni
d — 11 ;
and .*. n — x. (•.• d-x).
Second form.
J-x,
D — j, i.'.D-x),
Q — d, (.-. Q-x),
K — q, (.-. ^-a;),
Y — k, (.'. T-x),
Z— y, (.-. z-x\
z — n :
The argumentation is supposed, of course, to have
taken place on the lines of the process in the first form,
and the second included Sorites did not therefore come
into the process until the proposition, Y — d, was dis-
puted. The illustration thus shows the superiority of
the first over the second form, as exhibiting the whole
OO LOGIC AS A PUKE SCIENCE.
process of the reasoning. The second could not have
been framed until the first had been gone through with.
§ 24. Compound Sorites may, however, be exhibited
in forms which at first sight may seem to be in contra-
vention of what has been before laid down, but ujjon
examination it will be found that such is not the case.
Thus, in the two following cases : "^
(1.)
(3.)
■ N —
d, N - d,
D —
J, D - h
^ —
X, J — X,
Y —
X, ¥ - X,
Z —
y, z - y,
Q -
z; Q — z;
/. ^ —
Q. '.i^- Q.
Let us take the second and write in line with each
premise (except
the first and last) the implied con-
elusions :
N — d,
D — j, (.-. ]!f-j),
J — X, i.'.N-X),
Y — X, (. . ^- D,
Z — y, (.-. ^-z),
Q - z;
.'.
5^ - Q.
The expression, with the exception of the ultima, will
be found, upon examination, to constitute the premises
of two simple Sorites, of which the first is in the pro-
gressive configuration and the second in the regressive.
♦ Taken from Schuyler's Logic, p. 88.
SORITES.
87
By stating them successively with their implied
ultimas, we shall have them in the following fonn :
■
N— d,
1
D- J,
1
J — X ; (.-. N— X, held in the mind).
1
¥ — X,
i-
z — y^
1
Q — Z ; (.-. -^-X, held in the mind).
and then.
•/ N — X,
and ^ — X;
.-. ^-N;
or,
•' ^ - X,
and M — x;
/. N - Q.
I now proceed to show that Sorites, stated as above,
fall within the definition of compound Sorites, as herein-
before given.
The maximus premise, being the last premise oi the
principal Sorites involved in the foregoing examples,
has not appeared, but has in all cases been implied.
The middle premise is (as has been before stated to be
the case in all Sorites, simple and compound) the second,
and in the examples is D — j. Combining this with the
ultima, the abridged form of the principal Sorites is
therefore,
D- j;
/. ?f — Q.
e8 LOGIC AS A PURE SCIENCE.
Expanding this in the 12th progressive mood as in
the synopsis, we shall have the full principal Sorites as
follows :
N_d, D-j, J-Q; .'. ^-Q;
and the compound Sorites will be as follows :
N -
d,
D —
j;
J —
X,
¥ —
X
Z —
y.
Q-
z;
(.•. N—j, held in the mind).
X ; (••• ^— y, held in the mind).
^ - Q;
and .*. N — Q. ( .• n^-j).
§ 25. But the magnus and maximus terms of the
principal Sorites, at the ultima of which we first arrive,
may not be the infima species and summum genus, and
further investigation may bring into the process of the
reasoning lower species or higher genera, and if in both
directions, both ; and the new term or terms, instead of
being employed interiorly as middle terms as hitherto,
will be employed exteriorly. In such case the new
term, or terms, will constitute, if there be but one,
a new magnus, or maximus term ; or if there be two,
obtained by investigation in both directions, both, and
SORITES. 89
the displaced terms will become middle terms. We shall
then find that there will be two new abridged and full
principal Sorites in each case, one regressive and one
progressive, but varying according as the new term, or
terms, are applied to the original Sorites considered as
both regressive and progressive. They will, however, be
independent of each other, and each will have its correla-
tive in their respectively opposite configumtions. The
displaced original term, if it shall have been the magnus,
will become the minor-middle term of the new principal
progressive Sorites, and will not appear in the new re-
gressive, but if the magnus term be again displaced by
bringing in another, then the displaced original term will
become the major-middle ; but if the displaced original
term shall have been the maximus, then it will become the
major-middle term of the new principal regressive Sorites,
and will not appear in the new progressive, and if the
maximus term be again displaced by bringing in another,
the displaced original will become the minor-middle term.
But of the original premises in the case of one new
tenn being brought in, one, or two, wall still remain in
each new principal Sorites, one in the regressive configu-
ration, and two in the progressive, if the new term be
maximus, and vice versa, if magnus. One original premise
only will remain in each of the new principal Sorites in
any case if two new terms, one magnus and one max-
imus, are brought in.
The original ultima will of course have disappeared in
every case. But if two new terms are brought in, both
having been discovered in a process of investigation in
one direction only, the original ultima will reappear as a
•90 LOGIC AS A PUEE SCIENCE.
'premise of one of the new principal Sorites, the regres-
sive, if the investigation were in the ascending direction,
and the progressive, if in the descending.
If the investigation shall be pursued so that more than
two new terms shall be brought in, in each direction,
every vestige of the original principal Sorites will have
disappeared from the new principals, as they will then
be constituted.
But all the premises of the original principal Sorites
will, in all cases, be found to remain, either partly in the
principal Sorites, and partly in the following included
Enthymeme or Sorites, or in two of the included Sorites,
or wholly in the last included Sorites, or partly in the
Enthymeme, which is the final expression, and partly in
the next j)receding included Sorites, according as the
new terms shall be brought in ; and they will always be
found together in their original order, either regressive
or progressive, how far soever the process be continued,
and this, also, whether the compound Sorites be in the
first or second form, as hereinbefore shown.
The following examples illustrate all the foregoing
remarks, except the last, as to compound Sorites in the
second form, which can be verified by trial. All the in-
volved Syllogisms are in the first figure throughout.
The original premises and ultima (employed as a
premise) are printed in Roman letters, and those which
remain in the j)rincipal Sorites in full-faced type. All
other propositions are printed in Italics. The examples
having the same number of new terms are so arranged,
either on the same or opposite pages, that they may be
readily compared.
SORITES. 91
With one neio term, brought in in the ascending process of in-
vestigation, and therefore a new maximus term:
Regressive Configuration.
J _ X : (•• J-
-yh
Progressive ConfigurcUion.
N-d,
D — j ; (.-. N-j)
K — d;
* ' -J
J — X,
N- j,
and .-. X — y. {.■ J-
y)'
J — y,
and .-. N — y. (•.• n-J).
Full forms of neio principal Sorites :
^—y, J — X, iV^— y; .-. N—y.
X_cl, D — j, J — y. :. N — y.
With one neio term, brought in in the descending process, and
therefore a new magnns term:
Regressive Configuration.
Progressive Configuration.
J — X,
K — n,
D — j: i.-.D-
-ar).
X — d;(.-. JT-flO.
N- d,
D- j,
A' — n ;
J - x;
K— d\
I) — x\
and .-. K — x. (•.•/>-
x\
and
:. K — .r. c- K-d).
Full forms of new principal Sorites :
J — X, D — j, K—d\ .'. K—x.
K—n, N— d, D—x\ .-. K—x.
9^ LOGIC AS A PURE SCIENCE.
With tivo new terms, one brought in in the ascending process of
investigatiouy and therefore a neio maximus term, and the
other brought in in the descending process, and therefore
a new magnus term:
Regressive Configurat
ion.
Progressive Configuration.
X- y,
K — /^,
J — x; (.-.
J-
-y\
N — d; {.-.K-d),
D - j,
^ - j.
N — d.
J — X,
K — n\
^^-//;
,',
.
..
A'-i;
D - y.
and /. K — y. (•.•
J-
-y)-
and .
\ K — y. {,-K-d).
Full forms of neio principal Sorites :
X—y, J —X, £^— j ; ■
: K-y-
K — n, N — d, D—y; .
: K-y.
SORITES.
93
With two new terms, both brought in in the ascending process
of investigation, and 07ie therefore a new maximus term :
Regressive Configuration.
Y — z,
X — lj\{.-. X-z),
J — X,
D - J,
N — d;
y — x;
and .-. X — z. (•.• x—z).
Full forms of new principal Sorites :
ressive
Configuration.
N
- d,
D
— J; (-•
. iV^-
-J),
J
— X,
X
— y^
r
— Z'y
J
— ^
/. X
— Z. C
• N-
-J)-
Y- z,
X-y,
X— x:
.'. X—z.
N— cl,
D-j,
J — z:
.: X — z.
With two new terms, both brought in in the descending process^
and one therefore a neio magnus term:
and
\ive Configuration,
Progressive Configuratioii.
J- X,
Q-k,
B - j; (••■
D-
-ar),
K — n; (.-. $-n),
N — d,
N — d,
K — n,
D-J,
Q-k;
J — x;
Q-d;
X- x;
Q — ■^. (•••
D-
■x-).
and .
•. Q — X. {:■ §-»).
Full forms of new principal Sorites :
J— X, D — j, Q — d; .'. Q — x.
Q — k, K — n, N — X ; .-. Q — x.
94
LOGIC AS A PURE SCIENCE.
With three new terms, of which two are brought in in the ascend-
ing process of investigation, and one of them therefore a neio
maximus term, and the third in the descetiding process,
and therefore a 7iew magnus term:
Regressive Configuration.
r- z,
Progressive Configuration.
K — n,
N — d; i.'.K-d\
1 D
j ; i.-.D-x),
{.'. {:■ J)-x),
K — X',
and .*. K — z. c- x-2).
Full forms of new principal Sorites :
Y—z, X—y, K—x', .'. K—z,
SORITES.
95.
With three new terms, of which two are brought in iji the desce7icl-
i7ig process of investigation, and one of them therefore a new
magnus term, and the third in the ascending process, and i
therefore a new maximus term:
Regressive Configuratimh.
J — x; (.-. J-y\
J I) - J.
i N — (1; {.-.N-j),
K
Q
Q
and .-. Q
Progressive Configuration.
N-S).
J X,
{...
J — y\
(viV-i),
N — y;
and .-. Q — y. C- G-»X
Full forms of new principal Sorites :
X—y, J — X, Q — j',
Q-^lc, K—n, N—y,
Q-y-
Q-y-
96
LOGIC AS A PURE SCIENCE.
With four netv terms, of which two are brought in in the ascend-
ing process, and tivo in the descending :
Regressive Configuration.
Y — z,
X — y\ (.-. X- «),
Progressive Configuration.
Q -h
K — n\ (.-. q-n\
/ J - X,
N — d,
K — n,
f Q - d',
Q -x;
and .-. Q — z. c x- z).
J N-d,
I D — j; (.-. iVr-A
J —
x/
X —
y.
Y —
^;
_ _ -
- - ,
{...
J — z;
N — z;
and .-. Q — z. (.- Q — n).
Full forms of new principal Sorites :
Y- z, X-y, Q-x; .-. Q - z.
Q — h, K — n, N — z\ :. Q — z.
SORITES.
97
With eight neiu terms, of which four are brought in in the
ascending process, and four in the descending:
Progressive Configuration.
Regressive Configuration.
S — t,
Z — s\ (.-. z - 0,
I Y - z,
i . X — y\ (.-. x-z\
s J
3; (.-.V-x),
j X - d,
l K — n : (.-. K- d).
Q-
h'
G -
q^
H -
■9'^
—
— .
y .-. (V D-x),
i .*. (••• X- 2),
H — z\
and .-. H — i. ( .• z - 0.
(. K — n-. (.-. Q - «),
J N - d,
I D — j; (.-. ivr-i),
/ J - X,
i X — y; i.-.J-y),
r- z,^
Z — s,
S — t;
j r - t;
I .'. (••• J-y),
1
J —
t;
t.
(••
• A'-
-S).
{.
X —
t;
*.
(••
• Q-
-n),
Q -
t;
and .
.H —
t.
(•.
H-
-?)■
Full forms of new principal Sorites :
S-t, Z—s, H—z', .'. H — t.
H-g, G--q, Q-t', .-. H-t.
98
LOGIC AS A PUEE SCIENCE.
§ 36. To recur now to illustration by means of geo-
metrical figures.
A regular tetrahedron may by four sections, beginning
in the middle of each of its edges and made parallel to
the opposite planes respectively, be divided into five fig-
ures, of vrhich four v^ill be regular tetrahedra, and the
fifth and interior figure a regular octahedron.
Thus, by reproducing our former illustration on card-
board before folding, and dividing it by lines which
shall represent the four sections, we shall have the
following :
ISTow, assuming each interior dotted line to be the
edge of an equilateral triangular plane, represented by
card-board, projecting backward, divergingly, at the
proper dihedral angles, from the plane of the one which
we are supposed to have in hand, then, by folding the
latter as before, we shall have a combination of five
figures, as above stated, which will present to our eyes
successively, as we turn it about as before, the following
figures :
SORITES.
First Series.
N\ /X
Second Series.
Each of the four tetrahedra having one original exte-
rior point, and three visible and one invisible planes, will
be found to have that point marked with one of the let-
ters N, D, J, X on each visible plane ; the fifth figure,
the octahedron, having no original exterior point, and
four visible, and four invisible planes, will be found
marked on each visible plane vdth one of the numbers
1, 2, 3, 4. It is wholly included, and occupies all the
space, between the invisible planes of the four tetra-
hedra and planes connecting their visible planes, and
its volume is exactly equal to the sum of their vol-
umes ; and it may well be regarded as the analogue
of the ultima conclusio of the Sorites, of which the
abridged form is :
D-j; .-. N
X.
The analogy between a compound Sorites in which the
original principal Sorites shall remain the principal, and
100 LOGIC AS A PUEE SCIENCE.
a Sorites be interpolated as hereinbefore shown, and
a tetrahedron divided by sections as represented in the
foregoing illustration, cannot be exhibited as simply or
as clearly as that between a simple Sorites and a tetra-
hedron considered as a unit, as in our former illustration,
because the tetrahedron which is the analogue of the
included Sorites is involved in and forms an indistin-
guishable, but, as must be regarded, separate, part of
the included octahedron, having one of the visible planes
of the octahedron as its only visible face. Its invisible
faces cannot be brought to the surface in the following
figures,, but must be regarded as represented by the three
triangles by which its visible face is bounded, the ultimate
point of which will be found marked X in the figures.
Its ultimate point will not be the point X as shown in
the figures, but will lie in the perpendicular let fall from
the point N upon the opposite plane of the original
te-trahedron. We shall hereinafter find that perpendic-
ular to be part of one axis of a sphere produced by the
revolution of the tetrahedron, and that the pole of that
axis opposite N should be marked X. The ultimate
point of the indistinguishable tetrahedron which is the
analogue of the included Sorites, may be at any point
in the line of this axis within the octahedron, and let us
assume that point to be in the centre of the invisible
plane of the octahedron opposite its visible plane which
is the visible face of the involved tetrahedron. The in-
visible faces of the latter will then be equal to the tri-
angles by which its visible face is bounded in the figures.
Let us suppose that in the progressive process we
have established the relation between N and J, as in the
lower one of the following combination of triangles
SORITES. 101
(which, observe, are the same as the triangles 1 and 3
in our original card-board illustration), and that the
relation between J and X requires to be established.
We shall then have the upper triangle in which only the
relation (length of line) between D and J is known, and
let us suppose that the relation between each of those
l^oints and X is not capable of being immediately deter-
mined, but that there are two points {middle terms\ one
in each of the other two lines, capable of being succes-
sively reached from D or any point in the line B J ex-
cept the point J, and the length of a straight line con-
necting them capable of being measured, and from both
of which the direction of X can be observed, and the
angles therefore determined.
Keproducing the upper triangle and marking the mid-
dle point in the base line J', and the two points at the
extremities of the base X' and X", and the two new points
Y and Z at the middle of each of the two lines connect-
ing the extremes of the base with X, and connecting such
new points, and each of them with J', we shall have the
following :
102
LOGIC AS A PUEE SCIENCE.
and we may now establish that J' — X in the same man-
ner as we have hereinbefore established that N — X.
But, the lines X' X and X" X are, by construction,
equal to J X and D X in the upper triangle, on the
preceding page, the middle points in which may be
marked Z and Y. In the process we have found J' Z
equal to J' Y, and X' Z equal to J' Z. X' Z is therefore
equal to J' Y. But X' Z is J Z. And as J' Y is equal
to J' Z, it will, upon being applied to the latter, coincide
with it, and the point Y will fall upon the point Z.
J Z may therefore be called J Y, and is equal to J' Y.
The whole combination of triangles will now be as fol-
lows, the original letters being put on the outside :
SORITES. 103
We can now express the full compound Sorites, ex-
emplified by the foregoing illustration, as follows :
X - d,
D — j: {.'. X—j, held in the mind),
J — y. (= J Z = J' Y, fi-om which latter directum of X odserred),
Y — Z, <=T Z, relation, i. e., length of line known),
Z — X : {= Z X, direction observed from former),
J - X ;
and .'. X — X. (•.•x—j).
This is the same compound Sorites as that exhibited
in § 20, on page 81, but with the included Sorites in the
progressive, instead of the regressive, configuration.
But if the interpolated expression be an Enthymeme,
the analogy will be much clearer, as the lines by which
the Enthymeme will be represented wiU lie wholly in the
surface and not involve any section of the original figure.
Thus, if in the following combination of triangles
(which observe are the same as triangles 3 and 4 in our
original card -board illustration) :
we shall, in like manner as before, have established the
relation between N" and J (as in the upper, left-hand
triangle^ from which latter we can see X, but are
unable immediately to determine its distance, with-
104
LOGIC AS A PURE SCIENCE.
out the knowledge of which we cannot establish the
relation between N and X ; we may select another me-
diate point, Y, which can be reached, and distance
measured from J, and from which X may also be seen,
and the angles therefore determined, as in the following
figure :
and then, by the elements thus obtained, we can deter-
mine the required distance from J to X, and by means
thereof and the elements previously obtained, the dis-
tance from N" to X.
The compound Sorites exemplified by the foregoing
illustration will be as follows :
N — d,
D - j ; (.•
iV — j held in the mind),
Y- X,
J -y;
J -X,
.-. N — X. c
N-j).
But if, instead of having begun in the ascending
direction, we shall have begun in the descending, and
have established the relation between X and D, as in
the lower, right-hand one of the following combination of
triangles (1 and 2 in the figure on page 65) :
SORITES.
106.
and shall then, although able to see N from D, but not
from X, be unable to determine its distance from D,
without the knowledge of which, it would be impossible
to determine its distance from X ; we may, in like man-
ner as before, select another mediate point K, which can
be reached from D, and from which N can also be seen,
as in the following figure :
and then, as before, may determine the required distance
from D to N, and by means thereof and the elements
previously obtained, the distance from X to N.
The compound Sorites exemplified by the foregoing
illustration, will be as follows :
X comprehends J,
J comprehends D J (.
J) comprehends K,
K comprehends N J
X comprehends A ^d in the mind),
D comprehends N j
and .'. X comprehends N. {'.' X amiprehends B).
106 LOGIC AS A PURE SCIENCE.
By putting together the first of each of the two sets
of figures in the preceding illustrations, on the line D J,
common to both, we shall have the following figure :
which is the same as that on page 61, but in a different
position. By turning triangle 2 downward in a semi-
circle on the point D as a centre, we shall have our
original card-board figure ; or by turning triangle 4 up-
ward to the like extent on the point J as a centre, we
shall have the figure shown on page 65. Triangles 1 and 2
taken together and 3 and 4 taken together are analogues
of progressive Sorites, 1 and 2, in the descending direction,
and 3 and 4, in the ascending ; but if 2 and 4 be both
turned as above described, they will become analogues of
regressive Sorites in the respectively opposite directions.
§ 27. All the four triangles in our original card -board
illustration are equilateral and equal. The solid figure
resulting from the folding of the card-board is a regular
tetrahedron, which is defined as a solid having four
faces, all equal equilateral triangles. But the triangles
SORITES. 107
raight have been all equal isosceles triangles, or partly
equilateral and partly isosceles. Such can be exhibited
in a plane figure bounded, by three, or four exterior lines,
if the triangles are all equal, or by six, if they are partly
equilateral and partly isosceles, and capable of being
folded so that the exterior points shall meet in a perfect^
but not regular^ figure. But a 'perfect tetrahedron may
have all its faces unequal, and in such case the faces may
be spread out in an irregular plane figure having ^\q
exterior lines. In all cases the number of exterior lines
will be found to be six, if bisected lines are counted each
as two. All other plane figures having all the points
exterior are imperfect and cannot be folded, so that the
exterior points will meet, and their areas, and conse-
quently the volume of space which they can be made
resjiectively to inclose, can only be determined by means
of the triangle. Imperfect Syllogisms and Sorites in
logic must be reduced to the perfect figure before they
can be submitted to the dictum de omni et nullo.
§ 28. On the other hand, a tetrahedron (regular or
perfect) may be added to on the outside by superimpos-
ing on each of its faces another tetrahedron having a
similar face, so that there shall be ^ve tetrahedra in
all. Four new points vrill have been added, all exterior
to the original figure, the original x^oints becoming in-
terior, but their locations visible, the original figure
having otherwise wholly disappeared from view.
Similarly, as we have before seen, in respect to a
Sorites, when four new terms have been brought in ex-
teriorly, two in each direction, the four propositions of
108 LOGIC AS A PUKE SCIENCE.
the original principal Sorites will have disappeared from
the two new principals^ as they will then be constituted,
but they will remain in the included Sorites, of which
the inner tetrahedron is the analogue.
But in the figure, formed as above described, the four
new points, which we will consider as marked K, Q, Y,
and Z, will furnish only one new principal Sorites, as its
analogue, which may be rendered in its abridged form
thus y
V Q-y; /. K-z.
But observe, the interior figure in the foregoing com-
bination is a tetrahedron, not necessarily regular, but per-
fect ; and if, instead of beginning with such a one, con-
sidered as a unit, we begin with a regular one considered
as divided by four sections, as before shown, and super-
pose upon each of the visible planes of the included
octahedron, a tetrahedron similar to each of the four
resulting from such sections, we shall have a solid figure
in the form of an eight-pointed star, the octahedron
having entirely disappeared from view, except that the
locations of its points will be visible. This eight-pointed
star will be found to consist of two equal intervolved regu-
lar tetrahedra, to both of which the interior octahedron
will be common, and its revolution about its centre will
produce a sphere exactly equal to that produced by the
revolution of the original tetrahedron. Four exterior
points will have been added, but of these two are the
opposite poles of the two original points marked N and
X, and, having a common relation with them to the in-
cluded octahedron, should be marked X and N respect-
ively, leaving, in fact, but two new independent points,
SORITES. 109
which may be marked Y and Z. The whole figure will
then be the analogue of two independent full Sorites, of
which one only is new, and that only in part, the abridged
forms being :
•.• D — j ; .-. N — X.
*.• Y — z ; .-. N — X.
Bj' comparing the foregoing illustrations with the
Sorites having four new terms added exteriorly, given
on page 96, the superiority of the Sorites over its ana-
logue, the tetrahedron, will again be manifest.
§ 29. Thus everywhere, whether we go inwardly or
outwardly, and in all things, metaphysical as well as
physical, we find triniunity, and can thence proceed to
quadriunity, but beyond that, except in composite
forms, we cannot go.
§ 30. From the foregoing definitions and illustrations
of Sorites, simple and compound, it seems manifest that
the human mind is limited to reasoning concerning the
relations of four terms. If other terms are brought in,
they must relate to the terms of the principal argument,
and in such case, if such relation be to the middle terms,
they serve only to elucidate, but if to the magnus and
maximus terms, then they supplant those terms ; which,
if there be one, or two successively of each (new magnus
and maximus terms) respectively, become terms of the
two new middle premises respectively, but if more
than two of each, then are relegated to the subordi-
nate position of middle terms employed only in elucida-
tion. Otherwise they must be the terms of independent
arguments.
110 LOGIC AS A PURE SCIENCE.
§ 31. There remains but to say that I have not pointed
out the characteristics of Sorites, nor given the rules in
relation to them, as the same have been usually pointed
out and given (or in part so) in logical treatises, and to
which reference has been hereinbefore made ; and I now
refer to them only for the purpose of showing their
inadequacy.
They have been written with reference to Sorites
treated of as capable of being expanded only in Syllo-
gisms wholly in the first figure^ and without reference, of
course, to the distinction between them as simple and
compound, which has been hitherto unobserved. They
relate,
1st. To the number of Syllogisms involved, as equal
to the number of middle terms, and as ascertainable from
the number of premises of the Sorites, less one.
2d. To the character of the premises of the involved
Syllogisms, whether minor or major, and the number of
each and their sequence, viz.: one only, and that the
first, major, and all the following minor in a regressive
Sorites ; and mce versa^ in a progressive.
3d. To the number and positions of particular and
negative premises in the tw^o configurations, viz. : that
one only can be particular, and that the last, and one
only negative, and that the first, in a regressive Sorites ;
and vice versa (in respect to positions) in a progressive.
The first is true of all Sorites, simple and compound,
in respect to the number of Syllogisms involved being
equal to the number of middle terms, and has been im-
pliedly shown as true of all simple Sorites, in respect to
such number being ascertainable from the number of
SOillTES. Ill
premises less one, in that they have been described as
having three premises, and as being capable of expan-
sion into two Syllogisms ; but in such latter respect it
does not apply to compound Sorites when fully ex-
pressed.
The second, by an examination of the synopsis, will
be found to hold good, of all regressive simple Sorites in
respect to the moods in which they are minors, and not
good in respect to those in which they are majors, and
mce versa of all progressives.
The third is of course, and for obvious reasons, appli-
cable to all simple Sorites (but not to all compound^
when fully expressed)^ so far as the number of particu-
lar and negative premises is concerned, but to state it in
respect to their positions as applicable to all Sorites
capable of being expanded in Syllogisms ickolly in the
first figure^ and also to some in combinations of figures,
either partly or not at all of that figure^ and then to
point out the very numerous exceptions in other like
cases, would tend rather to confuse than to enlighten ;
and I therefore leave the subject, and pass on to the con-
sideration of Fallacies.
U^ 0? THE -<*-
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